Oh dear, a difficult situation. We had a similar situation with one of the coaches at our gym earlier this year. The child was petrified of a certain skill and yet the coaches felt the more they pushed her the more likely she would get it. But from my dealings with this child in the past it was obvious that this was not the case. She was the sort of child that if you pushed her it made things worse.
We worked with the coach to help him understand. Is there another coach at the gym who might understand these issues and be able to communicate them better with the head coach?
I would set up a meeting and explain that the strategies put in place are great (Make them feel good about themselves and like you respect them will put them on side) but that your daughter is a unique case (make them think again they are great coaches that its just your daughter thats the problem). And that a different strategy may be required. It seems that if this is the HC and there are no other gym options you may need to stroke his ego a bit to get him to work your way.
Ask your daughter to write a list of every stage she can think off between doing the double back handspring on floor to doing it on high beam. Go through everything from a line on the floor, foam beams, low beams, mats over beams, mats under beams and spotting etc. Have her tick off the ones she feels comfortable doing and the ones she does not. Have her go to the point on the list where she feels comfortable and work from there. Work slowly up, if she gets to a stage where it isn't working go back a step.
Encourage your daughter to recognize and ask for what she needs eg "can I get a mat and put it over the beam for the first few" or "do you mind if I do xxx drills first to get my eye in". The way she asks can make a big difference.
. My dd does the 1st way you describe the 'L' one.
