First I am not so sure if I would bet the flop to start with. Is checking behind horrible here?

As played and taking it as a given that this is his range:

1) if you have no fold equity at all (read: he stacks off with AQ 100% of the time), why shove with only 46% equity (b4 rake)? Note that you don't need very much fold equity to make the play profitable though (take rake into account when you calculate this).

2) you get 23.3% pot odds for a call, and you have 9 flush outs (all clean because you block) and 5 outs to two pairs/trips but these beat him only 9 times out of 16, so count 2.8 outs. Total 11.8 outs for about 21.5% chance to hit on the turn, which is less than your pot odds. Also, when you hit a flush on the turn your implied odds become very limited vs 9/16th of his range and are rather limited vs his sets since he knows (or should now) that he doesn't have many outs to a boat/quads. Furthermore, when you hit two pairs or trips, the turn card is never a club, so villain is likely to lead out with his entire range and you have to raise with 60 or 66% equity vs his range so all the money goes in.

So quick and dirty estimated EV of calling:
0.785*(-$3.75)+0.215*((9/11.8)($12.35)+(2.8/11.8)*(0.63*$27.25+0.37*(-$22))
=-$0.45

So in the end I would say it's a stretch between shoving or calling or folding?