After testing with another mod that depends on !skins, it also didn't apply the inventory changes. So it seems to be a compatibility problem between !skins (https://mods.factorio.com/mod/skins-factored) and the inventory mod.
Currently, the inventory mod modifies the character prototype, and !skins doesn't copy the inventory size data when switching between skins. I recommend forwarding the issue to both mod pages, since it could be fixed on either side. Thanks for the feedback tho :)