So, I have to say before I go any further that I am located in the northeast and things obviously operate a little differently here.

Different invasive species, different climate and different soil.

Some people will tell you that the grass will grow under there, and it will, but keeping it thick and healthy can be a problem. The seed mix you are using would be best at 5.9-6.3 pH whereas trees and woody shrubs prefer a pH closer to 7. The difference may not seem significant, but combine that with competition with the roots and it being shady(though fescues tend to thrive in shade if you get the right variety) and it could make things tough.

So it actually wold seem that possibly mulching under the tree could be more ideal. YOu want to be carefule not use the conventional red cedar mulch though as that affects soil health, nor do I recommend pine bark mulch bought at lumber yards etc. Your best bet is to find your local arborist and see what he has for wood chips, and use this as a mulch. In addition to this you really should avoid the current trend of mulch mounds around the root crown of trees as this can damage a tree(I wont give a long explanation as to why , but you should look it up this trend really should stop). The best option would be a thin layer of mulch for a wider radius around the tree, on which you could put some patio furniture etc…

Another idea is to plant some sort of shade loving ground cover, that is native to forested areas of NM. For this you can call your local extension office and find out what they recommend, I don't think that pennsylvania sedge will work like I had said initially as it likes the northeast climate most, but it may and your extension office will kow for certain.(An extension office is the local outreach to the community from agricultural and ecological services in your area, usually based out of a university)