+1 If you want to run an 11-36 cassette, it will have to be Shimano. So you'll need a Shimano rear wheel.

I have my road bikes setup with Campy 10 speed shifters and an 8-speed Shimano drivetrain. Works perfectly. To do this, you'd need a new rear wheel, Shimano RD, 8S cassette, and 8S chain. But you'd need all those things to run a 10S Shimano cassette anyway, plus the shiftmate.

Any chance you could borrow a Shimano equipped bike for this race? Or maybe try throwing a MTB triple on your current bike. Something like 22/32/42 should get you up that mountain.