Game Version: 1.19.2
Forge: 43.2.10
Cloak and Dagger: 1.3.5
Footwork: 2.6
Domestic Innovation: 1.6.1-1.19.2
Citadel: 2.1.3-1.19

Domestic Innovation adds the ability to disable friendly fire for pets. In other words, you cannot hit a pet unless you are crouching.

Issue: Occasionally, the pets are affected by the stealth system and may not properly follow the player, they will have question marks and other icons switching on top of them, and can also be accidentally hit even if you are visible in front of them with the open eye icon.

I'd imagine there is some internal tag that all tameable mobs use that could be applied to this mod to avoid this problem right out of the box, right?

In reference to the issue of #2 this problem can be fixed by adding pets in the "mobs" list on the config. Such as "minecraft:dog, ahmvod" and "minecraft:cat, ahmvod" in order for pets to not be affected by the stealth system. This is useful to add even without the domestic mod to prevent doing extra sneak or distracted damage to your own pets!