What I'm trying to do is remove the reins from the EA bridles. I successfully edited the basic bridle, no problems. But on the Show and Western versions, I am having issues with the bone assignments doing strange things.

TRY 1: I exported the mesh from TSRW to .obj format. I edited the mesh in blender (I'm more comfortable with doing the edits there), exported the edited mesh. Imported the new mesh into Milkshape and exported to .wso format. I used mesh toolkit to auto-assign the bones using the unedited version of the bridle as the reference. I don't have pictures of this version, but it was BAD.

Try 2: I exported the mesh from TSRW to .wso format and did the edits in milkshape. I very carefully deleted the reins and did not touch any of the rest of the bridle. I didn't touch the bone assignments anywhere AT ALL. The bone assignments should be the same as the original version. I even looked at the original with the vertices colored with the bone assignments before and after and the colors looked the same to me. Everything looked fine in TSRW (not that it means much diagnostically I've found...) For the BASIC bridle, this worked just fine. Success! So I did the other 2 bridles. But for the other 2 something went wrong. The western one is the "less bad" of the two. The Western bridle deformed slightly on the right side only (the section right below the concho decoration by the ear and a little towards the muzzle). The Show bridle did bizarre things. I have repeated the procedure a couple of time - and I am VERY careful not to touch anywhere near where these anomalies are occurring.

The only difference I found is that the basic bridle only had 1 mesh group. The Show and Western had 2 groups. For the Western Bridle, one of the groups was contained entirely in the section I was deleting. For the Show Bridle, I regrouped both parts into 1 mesh and then deleted out the bits I didn't want. These are the only differences in procedure between the one that worked and the 2 that had strange results.

So I figured out how to view bone assignments in TSRW and I looked at the ORIGINAL bridle. Nothing weird AT ALL about the bone assignments except maybe a few are under-weighted by 1 or 2 %.

BUT... When I look at the version I exported using TSRW and imported into milkshape - many of the bone assignments are different. And it's just certain bones here and there. I have tried using older version 2.0.86 version and 2.0.88 (because these were recommended particularly for clothing) and I also tried the most current version to do export. For each export test I did, I installed the comparable MS3D plugin for that version of TSRW to import into milkshape. In all 3 cases, the bone assignments got changed. What I've noticed is that the same group of vertices are affected each time.

So I don’t know if it is happening when I export from TSRW or if it is happening when I import into milkshape, but it is consistently happening.

But at least now I know what the bones are all supposed to be so I can (tediously) fix them.

I'm not familiar with the TSR Workshop workflow, but my Blender add-on can import GEOM files directly extracted from .package files and supports rigging (Horse rigs included ).
I can't say for sure if it works for Blender 2.9x, you'll have to try that, but it should definitely work for 2.8x

As for the bone weights, I seem to remember that TS3 wants all the weights normalized (a.k.a. all assignments combined must add up to 1.0) or it can do things like in your pictures.
You also can't have more than 4 assignments per vertex, but the exporters in Milkshape/Blender should take care of that.

The reason I am doing it the way I am, is because I'm extremely new at this and the handful of tutorials I have seen for frankenmeshing have you mess with the bone assignments in milkshape and/or mehstoolkit. For whatever somewhere between TSRW and Milkshape the bone assignments on these specific meshes get garbled. I honestly had no idea there was any other way to do it... but I surmised from your suggestion that there was. {{facepalm}}

TBH I would much rather do my editing in Blender. It is soooo much easier (to me) to use and control. So after a session of random poking of buttons and banging my head on the keyboard a few times in the google search box... I think I pieced together how to do what I need to do. I might not even need to dork with the bone assignments, given that your tool will import/export with them (presumably) intact. But at least I know how to do it now.

EDIT: I spoke too soon... I'm still flailing around. I just don't know enough about how to do this kind of edit going down this path. I understand the steps to do the way I was doing it except that either the import into milkshape or the export from TSRW is garbling the bone assignments on these specific meshes. I just don't know how to work with the geoms. I managed to get the geom out of the package and into blender and do edits and figure out how to see the bone assignments (and verify they aren't mangled). but sometimes i get errors trying to export it. And then I'm not entirely sure what to do with it after that.

Okay so I dug around and found an ancient tute on frankenmeshing that still uses milkshape, but talks about using s3oc/s3pe and importing export geoms. So I figure I could cobble together that knowledge with what I was able to glean about using SmugTomato's blender add-on. I managed to get the bridle clones, extracted as a geom and imported into blender. I figured out how to see the bone assignments and confirm that they did not get mangled like they were when exporting from TSRW and importing into Milkshape. So I managed to edit the mesh to my liking. The one wrinkle is that the mesh is in 2 groups. For the show bridle, I can keep it split but I don't know that it is strictly necessary? For the western though, one whole group is made up of the section I'm deleting. This doesn't seem like it should be a problem? In theory I would think I could clone the basic bridle, which is one group and replace it with the western mesh and textures?

HOWEVER.... When I go to export the mesh using the blender plugin, it appears to export just fine, but when I try to replace the mesh in the .package file, I get a mass of errors.

So I thought that maybe I needed to renumber vertices like the frankenmeshing tute said. So I did that - but the question was, what is the starting point for bridles? So someone suggested I use meshtoolkit to examine the a bridle and see where they start. So I did that and bridles don't appears to have vertex ids?? I even opened a regular piece of clothing to compare - and I see where in the clothing it has vertex ids, but not the bridle. I threw caution to the wind and tried renumbering anyway... and I started at a few different places... All ended in failure.

I don't know anyone who does pet parts to ask...

Could you upload both the original package(with the unedited GEOM files) and the edited GEOM so I can have a look?
I don't know why it would give you an error in s3pe, unfortunately I don't have any experience working on pets parts though.
As for the vertex IDs, I don't know if my add-on can even deal with something not having Vertex IDs at the moment, I'll have to review the code to make sure.

Sure can. I am including 4 groups here:

ahBridle - this is the basic bridle. It's mesh is a single piece. I can import it, edit and export successfully. When I replace the geom in the package with the edit geom I do not get and error. I tried this version in game and other than some slight variations in shadowing on it, it looks good. Also for note - this bridle worked fine in the TSRW/milkshape trials. It's bone assignments did not get borked. Both of the other 2 bridles' bone assignments get messed up somewhere between exporting from TSRW and importing into milkshape (prior to any edits I do).

ahBridleWestern - This bridle's mesh comes in 2 pieces. All of the mesh contained in lod1-1 is part of the section I want to get rid of. Essentially, I just deleted that group and then edited lod1 to remove the rest of the reins. This mesh won't even export using your tool.

ahBridleShow_2pieces - This bridle's mesh is also in 2 pieces, but part of the mesh in lod1-1 is a section I want to keep, so the resulting mesh is still in 2 pieces. These parts will export just fine, but when I replace the geom in the package, I get the error.

ahBridleShow_1piece - This version, I joined the 2 pieces into 1 mesh. This version will export. I tried to import it over a clone of the ahBridle that is only 1 piece, and I get the same errors as the split version.

That's weird, I can export them all just fine from Blender, but I am getting the error in s3pe.
Have you tried testing them in-game despite the error S3PE is throwing you?

Okay, I put the darned things away overnight. When I came back to them today, all I did was just re-open the files, no edits or anything, and magically they exported no issues. The geoms still show an error in s3pe, but I went ahead and tried them in game like you suggested. They are finally working! WHEW! With this method I didn't have the wild bone re-assignments like what happened using TSRW.

I'm glad you managed to get it working!
I'm also very glad I found out not all GEOM files have the vertexID attribute, it turns out the add-on already handles that properly, but I can't remember if that is intentional or accidental.
Turns out the code behind it was actually decent enough, so all I'll have to do is add a simple button to delete ids if needed.

This doesn't technically answer the question of why TSRW/Milkshape was mangling these meshes in the first place BUT you gave me a MUCH BETTER workaround! Thanks for the assist! I am so glad to have learned how to do bone weights in blender, though I didn't actually need to do anything than review them in this project. It will come in hand for other projects.