Yes, the true risk in tanning is body temperature. The problem is, the 10 minutes or so that you might be able to spend in a bed normally without heating up might be too long when you are preggers. Your hormones are different, body temp is different, skin is different, etc. I have a friend with 3 kids who is a nurse, and she said if I wanted I could tan, but just go in for a few minutes. What it boils down to for me is that it isn't even worth my time. I agree with the fake tan, maybe a little bronzer, etc. That is probably the best thing to do. Do I think you will hurt your baby if you tan once or twice, no. Might it dehydrate you quickly and make you feel ill, yes.
I used to work in a salon. One of the girls who was pregnent would go in and sit next to the bed and put her legs in...LOL. She felt it would make wearing shorts more bearable. I have a face lamp I use at home, some. It helps me tremendously when I feel pasty. Adds a little color, but no danger to body temp or baby. That's another thing you might consider.