In order to make a Fapet follow you, you first need to start a Hact with said Fapet (interact to have sex).
After this, hold the number 0 to stop the Hact (a withe bar will appear and must be filled). This way the Bunny will count as follower, and sleeping on the bed should be available as part of the script.
If you still cannot sleep on the bed, try pressing F5 and try again after getting a little far from it, otherwise, try restarting the game.
And, to see if you have a "Fapet Follower" look under your Stamina Counter, it appears as a strong yellow circle with a number. Also, the carrot chasing is just part of a Sex interaction, not the actual following