I like starbound in all, but this new ship pets thing is a bit annoying. You always have to look after it, and getting carrots (in my case, since I'm an Avian). I wanted to work on my ship but this pet keeps getting in the way.
You could put his/her food in a small room of your ship and trap him there. Then you can build in peace. (BTW, My bunny loves Thorn Fruit. It's easy to get and plentiful. You're bunny won't go hungry for hours. That way, you don't need to worry about feeding him too much.)
it's seem it does'nt work, the pet can escape the room when you are not in the ship (on a planet/outpost) tested on tier 4(and more) ship, with and without npc on board
Take it out on a date. Find out what it likes, and then what it dislikes. Then start doing everything it doesn't like. This is guaranteed to turn off your pets. I recommend the Star Cafe, it's over in what used to be Sector X.