Hi Mark, as you can see there are gyros out there that will perform at alt and gyro pilots who do fly at those sort of elevations.
Not sure it will be easy finding an instructor who instructs at those sort of elevations which is a little bit of a bummer as mountain flying does require additional skills whether it be fixed wing or rotary that you are doing.
It is possible, just a little more difficult for you than the average guy who wants to get into gyros. If you do go and get instruction, chances are that it will be at lower elevations, then when you go back home you will be operating just that bit closer to the edge of a standard gyros envelope.
They are not normally known for their sparkling performance or good climb rate, however there are machine out there that can perform well at altitude, but they will be more expensive than the average machine as the engine cost and rotor cost will be greater.
If cost is no factor, and it generally is, go and get your Instruction, at as high an elevation as you can find, You don't have any previous experience listed on your profile just indicate you are new to gyros but when you have the licence and machine, probably a good idea to get some mountain flying instruction in a fixed wing in the area you will be flying your gyro in unless you already have that experience..
The gentleman who springs to mind when talking of light aircraft and mountain flying is Steve Fosset. A man who held multiple aviation records and extremely experienced pilot killed in a Super Decathelon while mountain flying. Informed speculation, the conditions he flew into exceeded the performance of the aircraft and it was CFIT (controlled flight into terrain). The cautionary moral of that being that as good as any one is, operating at the outer edges of the envelope is more dangerous, and you unfortunately are geographically are towards the outer edge.
I'm impressed at 26,000' for the MTO, getting any aircraft towards the limits takes skill and effort.