I can only share my experience and will refrain from advice since what works for one, may not for another.
Last year, my DD was held back because of beam. She could not tackle the BHS in her brain......Stress was mounting in the gym..........At the right time, not before/during/after practice, we talked about it very calmly. She told me she had made her mind up that she was not ready for it and she was fine if it held her back a level. After assuring her that I was proud of her and how far she has come in her 11 years, we talked about it scientifically.....We both concluded that her brain had not yet 'wired' the skill and that she needed more repetition. The next problem for her was how NOT to upset the coach when the moment came to work on the skill in practice.....the coach, not knowing what my dd was going through, was getting frustrated with her. I told my DD she had to TALK to her coach and explain her fears (kids hate talking to their coaches...). Then I asked my dd for a few alternatives to make the coach happy in case communication was not possible......could she do the skill on the floor, on a mat, on a line, on the low beam?...etc. She was positive she could do it on the floor on a line.....So , she took it from there- I told her the only thing she could do was start somewhere, and work up.....little by little.....show her coach by working....and RELAX mentally... she gave herself SMALL goals.....to do it on the beam, with stacked mats, with the soft mat over everything- then slowly take things away. She would work from the floor line at the beginning of every beam rotation.... she was too busy working HER process to notice which girls were getting whatever skill........ She is 11 now and she has much better communication with her coach and she explains to her what her brain is doing. I dont fault the coach a bit....most kids are petrified to 'tell' the coach anything 'bad'.
She eventually got over the fear, and she learned how to come up with a plan (which I felt was most important) for progress. The coach was happy that she was working a solution without sitting idle and whining on the side.....at times, the coach would help her and give her a spot and then congratulate her.....
Now, I hope for her next fear, she can be equally calculating!!!!