Without this mod:

With this mod (gets also shifted with normal MC texturepack - in this case I'm using BDcraft):

Seems to be also an incompatibility with the Better Animations Collection Mod, the different lie down animations didn't get triggert and when deactivating the Better Animations Collection Mod, the texture-shift issue is gone.

There exist a fork from this mod, which didn't has the texture-shift issue. The other incompatibility is there still present too.
https://www.curseforge.com/minecraft/mc-mods/let-sleeping-dogs-lie-fabric

Just adding on to this -- I'm experiencing this issue alongside the Pet Armor mod by ViktorP04. I've tested with just these two (and their dependencies), no resourcepacks or other mods whatsoever. The wolf texture and model itself is fine regardless, but the model of the armor's middle segment gets shifted to the right.

(these particular screenshots were taken within a modpack)

Did you find a way to sonve it?

Sorry for the delayed response.

My Fabric port of this mod was a heavy refactor/rewrite of how the mod works, using Mixins to modify the wolf model directly rather than supplying its own, which is likely the cause of this issue.

Unfortunately I have no plans on implementing compatibility with other mods at this point in time, even less so for mods that are huge model/animation overhauls such as the Better Animations Collection.

I will leave this open, but if anyone wants to take a shot at adding proper compatibility, feel free to submit a pull request.

Hi ! Having the same issue for forge with critter armory mod (1.20.1), the armor model follows the dog position, but there is also a shift to the right.
Hope it can get fixed ! (Already opened an issue on critter armory's github).