I'm curious, what is your living elevation? When you climb 5000+ feet in a day are you traveling to a higher elevation? And if so, after the movement are you staying at that higher elevation?
I am not a doctor. But I ascend a minimum of 20K feet per month via the steepest trails I can find. This usually includes a movement of up to 7500 feet of gain in a few hours. I've never seen my resting HR affected and do not recall having seeing that happen to anyone doing the same work unless they lived at lower elevation and were visiting.
If you're coming to an elevation much higher than your living elevation, doing a movement, and then staying at a higher elevation, then your elevated RHR is most likely due to the body's response to the reduced partial pressure of oxygen.
But if you're doing a move and returning to your "home" elevation and your RHR stays up...again I'm not a doctor in any way but I'd really encourage you to talk to one.