If he does not want to compete pull him out, if he does want to compete, let him compete. Do not worry about the other gym at all. Unless they are in exactly the same rotation, he won't be watching them and they won't be watching him. You will be in the audience, and he will be just another little boy in another gym's competition gear. And if he is competing a lower level he is less likely to stand out and be noticable amongst bigger boys.
Also, I think you are either projecting your own worries, or having too many grown up conversations in front of him. Five year olds, even very bright, quite anxious ones, decide when to worry by taking their cues from the adults around them. If you are excited and happy he will see it as an exciting happy occasion. If you are worried and ambivalemt, he will pick up on that.
I know you have had gym problems and things are tricky, but your child is five. Five is very, very, very young. Everything he does with his spare time should be about fun and discovery and learning and enjoyment. His gym appears to be a huge source of stress and worry for you, and by extension for him. You have changed gyms and are still tense and anxious about things. Please consider some sort of circuit breaker. Rec gymnastics, watching less, a season off. He should be having fun and you should be having fun watching him. If this isn't happening, something is not OK.