Minecraft version: 1.20.1
Modloader: Forge / Fabric
Fabric loader version: 0.16.9
Environment: Multiplayer

First mod name: Follower's Teleport Too
First mod version: 1.20.1 - 2.6 Fabric

Second mod name: Prometheus (Permissions & Utilities)
Second mod version: 1.2.5 Fabric

I'm not exactly sure why, but it seems that the /home function used in Prometheus does not allow things like dogs to teleport with you. I've tested a little bit with varying distances, and it seems if you use /home under a certain distance then Follower's Teleport Too will still kick in; however, in-and-around 1000 blocks + it does not and your dog is left wherever you teleported from.

For some reason this issue only happens with the /home command and not the /tpa command that Prometheus utilizes.

Prometheus also has a /warp command, but I did not test to see if Follower's Teleport Too works with this command.