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Not certain your troubles. You have the interior door panels off? Correct? You have loosened the glass lower stop so you can run the glass down extra far?
you do own a set of low profile sockets?
Not certain what else.
K,but still need an access point to put nuts and tighten
I was of the impression 77 was the first year for the sport mirror option so should be two access holes in your inner door frame... frankly im not sure how those 77 sport mirrors were mounted, I can only assume they were mounted like any other corvette sport mirrors from the inside with nuts on the mirror studs.
GM F body sport mirrors were mounted from the outside with a bracket first and perhaps the same with 77? I wouldn't think so with the fiberglass door skin.
someone with more "77" experience would be better to answer
I lower glass. Remove the window outer felt/wiper strip. Reach through the top opening with nut in finger tips and install on stud. Reach in with 7/16ths low profile snap on socket with special low profile ratchet. Tighten nuts.