I'd try to finesse it a little on the shifting when standing, practice on flat areas? The only time I ever snapped a chain was on my hybrid mashing across an intersection. Wasn't much drama but I can hold my upper body up pretty well with my arms.

Edited to say I've also popped my rear wheel out of the drops mashing hard, but only once and probably because I wasn't careful after I had swapped on and off a kid trailer hitch mount.