I have no recent experience with them, but when I was into the elk hunting addiction the Jicarilla reservation in northern New Mexico was a great option. Godawgs57 is a serious elk hunter with recent experience. Horses sound great but they introduce many expenses and, ironically, limitations.
The second you include “lodge” in the equation (and there’s nothing wrong with that) ten grand goes out the window.
Elk hunting, it seems, is a definite “you get what you pay for “ situation these days. If 10K is your limit, then so be it. Have the very best 10K hunt you can, and set your realistic expectations accordingly.
Wishing you the very best of luck this fall.
