When an item with a custom model is worn on the head, it is visible in the first person mode

The first person view here does not look like First Person Model mod.

It just has vanilla hands activated.

Gotcha

As far as I know, these are not items in the head slot, but instead using armor stands or so. Nothing I can do about this.

That doesn't seem to be the case, I can recreate this on a local server with the following command and resource pack

/item replace entity @p armor.head with feather{CustomModelData:1005006}

smpo.zip

Then you have some conflicting mod installed. Head items get hidden correctly(otherwise player heads would also obstruct the camera):

It seems the FastAnim mod is making the model visible

Yea, FastAnime also had conflicts with other mods of mine. I'd suggest only using it when you dont have any mods that try to do stuff with the player model.