I share some of your same issues and here is what I do --
Burpees:
I go down and just stay down instead of continuing to go down and then get back up for a series of repititions; while I'm down I either do a series of pushups or regular mountain climbers.
Spider-man mountain climbers:
I work the obliques using that one move where we stay in a plank position and then lift the knee out and up to elbow. This really feels completely different than the spider-man mountain climbers because my pelvs/lower back is staying stable. This is probably the same reason why regular mountain climbers don't bother me.
As for the Cross-body kicks:
I work a completely different area of my body... the glutes. I go down to the plank position and then lift one leg up for a glute squeeze. I alternate legs to keep the same timing as the Cross-body kicks.
However, simply alternating side planks might be another good substitute.