I don't consider myself a mountain goat, yet, but here is something I have been doing to combat what you described. (And, I'd be curious if any experienced climbers think this is a bad idea for any reason.)

I am getting fitter and fitter for prolonged climbs because I am blessed with a mountain in my back yard. But, maybe this would work for repeats, too?

In order to train myself to keep going when either my heart or my legs want to explode, I have found I can take a sort of "rolling rest" on the bike, even uphill, by SERIOUSLY slowing down. Slowing both cadence and speed...right down to about 3mph at 60rpm in my granniest gear. It takes concentration not to fall over, but I find I can give both my heart and legs a rest for a couple minutes, and then I can gear up a bit and keep going.

I don't know if this is applicable on hill repeats, when you get a break on the downhills. But, it is something I have noticed helps me stop less frequently on hard climbs, for what it's worth.