I did that same trip and area the second week of June last year and was on a rented KTM990. I think your skillsets will play an important part of how you will be able to handle the passes more than the type of bike you ride. You've seen some very varied answers and bikes of different capabilities. A mediocre rider on the best equipment available will not do well and a well seasoned rider with skills can do them on any motorcycle. This was my first time in the area and also my first time on a big ADV bike and I feel that my moto-x skills played a huge advantage to handling the big bike. My riding partner was on a much smaller bike, KTM 690, and was struggling due to the altitude and lesser skillsets. I know of riders with years of road riding experience that have never gone offroad and quickly learn that offroad riding is a whole new ballgame. I've also seen MSF instructors get spanked pretty hard when confronted with dirt & rocks.
Attitude plays a huge dividend also if you are comfortable on your bike and you can handle it competently you will be ahead of the game.
It's been said here a few times already, but take it slow and methodical and you will have a blast.
I did Engineer Pass solo when my riding buddy gave up and turned around and went back to Lake City and rode the hardball around to meet me in Ouray. I give him full credit for admitting when he was over his head instead of becoming a liability to me.
Pic of me at Engineer's Pass