Go with my read or pick a better spot?

20 player NLHE home tourney with friends, $20 buyin, 3000 chips to start, WSOP blind structure, 20 minute rounds.

I made it heads up against a friend I invited, who is fairly new and was very lucky to make it that far. He hit a 4 outer and a 5 outer on the river to survive, and is calling raises liberally, especially when he's in the blinds. This was his first tournament with more than 9 players and more than a $5 buyin. A few weeks prior to the tourney, I lent him some books to read: HEFAP, SSHE, HoHE vol.1 (he didn't finish that though). He learned a few concepts, but has not had much playing experience.

When we started heads up, blinds were at 1200/600/200 ante. He had about 40,000 in chips, and I was around 20,000. I was planning to win small pots and chip away at him. However, he began to play somewhat aggressive, throwing in a pot sized bet or even a slight overbet on every flop. I was folding and I felt that enticed him to continue throwing in bets whether he had a hand or not. I was ready to take a big stand the next time I hit a decent flop. That leads to this hand:

Blinds: 1200/600/200 ante
I raise to 3000 on the button with 7-6o and he calls. I raised with the intention to steal the pot on a scary flop (which I've done a few times against him) or possibly hit a good flop and reraise when he predictably leads into me.

Flop: Qc6c3c
I hit 2nd pair, no club draw. He bets 6000. I'm 80% sure I have the best hand. However, my hand is very vulnerable and he will get a little less than 3 to 1 if I push. I feel that I'm a much better player, but I'll be shortstacked about 3:1, with less than 15 big blinds remaining.

Should I fold this and hope for a better spot?
Can I just call and push on a non threatening turn (Queen or non-club 7-2)?
Should I risk pushing this vulnerable hand even though I think it's best?