He's almost never folding 6th I imagine. I would probably just check behind, even with the straight outs and trip outs, we are crushed if he actually has Ks up or a big pocket pair:
http://twodimes.net/h/?z=2524072
pokenum -7s ks 3d kc 5c 7s 7d - 9h 4c 9d jh tc qs / 2d 6d 6s 6h td 2c
7-card Stud Hi: 1122 enumerated outcomes
cards win %win lose %lose tie %tie EV
Ks 7s Kc 5c 7d 3d 888 79.14 234 20.86 0 0.00 0.791
Qs Tc 4c 9d Jh 9h 234 20.86 888 79.14 0 0.00 0.209
has villain been raising a completion with junk, or just completing with a high card showing? If he's on a 4 flush, we are a bit ahead 58/42, but even if he has a low two pair like 44K to start, we are behind 54/46....basically as we are only ahead against a flush draw or just the bare 7s I like a check on 6th. Given that he reraised your completion then called the raise on 5th you have to give him credit for something right?