I would imagine an RV-14 would do fine here, I have a friend with one and have never heard him complain. However; high density altitude and mountain flying is often more about the pilot than the airplane. Every summer, we see accidents that did not need to happen. Most of them due to pilots that are not prepared and apparently too smart to listen/learn. Last week, a canard flyer decided to roll his airplane up into a ball because he didn't have a clue what density altitude is and ignored good advice.
You may already be aware... if so, please ignore. Before making such a trip, please familiarize yourself with what you will be experiencing.
MountainFlying.com has some good rules of thumb. Departing at 10,000 density altitude in my RV-4, I expect to use 300% the Vans published takeoff distance. Even after breaking ground, things happen (climb) very slowly (in comparison to low density altitude). Are you familiar with leaning prior to starting the takeoff roll? Are you aware that your indicated airspeed will be approximately 16% lower than at sea level? Are you aware that you will be taking off with ~65% power. Are you prepared for what looks like the ground rushing by at 300 kts but the airplane is still not flying?
Expect to be done flying by about 10am due to density altitude and ALSO wind/thermals/turbulence. The winds aloft forecast at 18,000' is VERY important. If the wind is above about 30kts at 18,000, expect it to be bumpy. If it much more than that, be careful.
Hmm... after proof reading that, it sounds very aggressive/in-your-face. I apologize, I don't mean it that way. I'm actually trying to be helpful but I'm just frustrated by the continuous stream of accidents and stories of people that scared themselves.