OK, the Cinch sockets provide a top surface about 1/2" above the PC card so the ones you found that say 7/16" inch high will work.
To remove the old sockets without destroying the plated through holes they are mounted in requires cutting each socket pin close to the socket itself and then removing them one by one. You will need a solder sucker or lots and lots of solder wick or a supply of round wood toothpicks to remove the solder from the opened socket holes. You will need 1/4" 6-32 threaded hex spacers under the boards replacing the 1/2" spacers now used. Run a 1/2" long 6-32 machine screw up from the bottom and screw the spacer down tightly on the top side of the chassis bottom, then drop the board on the protruding screws and fasten from the top with 6/32 nuts. Don't smash any wiring!
Then the 6CG7 tubes should fit with the cover in place.
Regards,
Frank
PS There are seven whoops, eight spacers under the board that will need to be replaced. Use Digi-Key part 2208K-ND.