My only input, having been to the top of both, is that there is nothing except a vault toilet at the top of Mt Evans, which also means no gates or fees, whereas the top of Pikes Peak has a gift shop & restaurant where you can get a DBR and is a more pleasant drive, imho, though has the potential for traffic, is gated and fees are required to go to the top. Both times I've been to the top of Mt Evans I have experienced headaches from the altitude. That has never happened to me going to the top of Pikes Peak, which I've done several times via motorcycle. IIRC, PP is a 2 hour up and back process, if you don't spend much time at the top.
Quoting Ira w/o context doesn't really provide info. ALL IBA rides start and finish with a DBR. That said, there is no DBR possible at the top of Mt Evans, so unless a Spot OK is acceptable as a start time, you're going to have to get a receipt at the bottom somewhere, then ride up to the top and ping the spot to show you were there, then head down.
The rules for the H2H only say:
This ride is from the top of Mt. Evans or Pikes Peak (the highest point in the continental United States) to Badwater, Death Valley, CA (the lowest point in the United States) via Independence Pass and Ely, NV.
Pikes Peak solves the DBR issue, but limits you to their hours of operation and as said, has the potential for traffic. I know that people have used Mt Evans in the past. You just need to factor that extra time in to your ride.