My advice-trust the coaches. He wants to compete this year, but doesn't need to compete this year. Stressing and pushing skills near or during competition season has only ever caused serious issues for my oldest son, resulting in long term blocks and frustration.
Tell him his old gym was unusual in that they would even let a five year old compete and most kids (boys especially) in T and T don't start competing until they are much older (and he would be too young to even compete at all this year in men's gymnastics).
Tell him it doesn't matter what level he competes or what skills he can do-he just needs to have fun, try his best and believe in himself. If they have him compete level 1 or 2 because he wants to compete so badly but those are the levels where he can be successful without a spot, be truly and genuinely happy for him and encourage him to go do his best at that level - don't push him to get to 3 or 4. There is no mobility score until you get to level 5, so backing off and letting him compete low levels as a confidence builder has no consequence-he could jump to level 4 or 5 next season if he is ready-he doesn't have to score out of any of the levels in between.
Finally, and most importantly, tell him every day that he did a great job in practice today, that you believe in him, and you love watching him do trampoline. If and only if he mentions needing the spot, tell him that you are not worried, his coaches are not worried, he is getting better every day and you know he will get it when he's ready. Then change the subject.
From past posts on here it seems like he is a very intuitive little boy. I believe that if you can truly and genuinely not care about what level he is or what skills he is doing and just care that he is having fun doing something that he loves, he will sense that and back off of his own expectations for himself a bit, which will in turn remove some of the stress and anxiety he is feeling about doing these skills. That, in turn will allow him to feel more confident about trying them on his own.