Your Fukien tea is already not looking entirely healthy - a number of leaves showing yellowing. I don't know how long you have had it, to know whether this is from what you have been doing or it was that way when you bought it.
In general, like most trees, it would benefit from being outdoors in the growing season, until the night time temps start going below 45. However, it is a tree. Trees are adapted to staying in one place. I think finding a good indoor spot and leaving it is better than constant in and out. When you bring a plant out from indoors, generally you have to harden it off, get it used to being out doors gradually. You can't do that if it is going back and forth all the time.
Your Fukien tea can handle being indoors all the time. It will need extra humidity (indoor, air conditioned air is dry) such as a humidity tray and daily misting. And it will need plenty of light. A dedicated lamp just a few inches away should work, but it is nice if it can be near a window, to get some natural light as well. The lamp should be on anywhere from 12 to 16 hrs a day.