To me it seems your only viable option is to use the Border near Koch Agach. Traveling to Tuva is gruesome, international bordercrossings between russia and mongolia are few and far in between. Also keep in mind that on the russian side there is a 50 km border control zone where you are not allowed to travel without permit – except you hold a valid visa and are on your way to or from a border crossing on a main route.
Crossing the border by road does seem to be an option, but only at certain border crossing points. (The crossing specified by Google Maps, unfortunately, seems to be sandwiched between the two ones open to foreign nationals.) There are some reports that you can either get permission or “talk your way across” if you’re feeling up to the task (and possible rejection). If you want to make totally sure, maybe contact an embassy in the area?
Mörön does have an airport, but there are only flights to Ulaanbator. There are no direct flights from there to Abakan, so you’d have to hop over to a major Russian city and then back to Abakan. Basically, it would be in your interest to find a land route.
Credit:stackoverflow.com‘