AQ with rags on flop

[Shorty35]

What is the correct play of this hypothetical hand (which seems to hit me about once every two games)?

$30SNG, 5 players left, blinds at 50/100.
player 1: T4000 (SB)
player 2: T2000 (BB)
player 3: T1500
player 4: T1500
player 5: T1000 (Button)

Assume that Hero is player 4 and is second to act with AQ after player 3 folds. Hero raises to T300, player 5 calls, SB folds and BB calls.

Flop comes 8,6,3 rainbow.

What is hero's move? Same with AK?
Firing another bullet seems to put my stack at an unacceptably low position; but checking seems to concede the hand.

[Marcotte]

Well, if BB checks, I would probably push on the flop. But 15 of my first 100 were 5th place finishes, so I'm guessing that's not the right answer.

At the $11 games, your push gets called a lot. If you check, Button has to go all-in or check, giving you a free shot at your two overs. If Button bets, you get to see what BB does. If BB pushes, you gotta lay it down. Even if he calls I think you have to fold, although it does give you a little overlay which I am too drunk to calculate.

[SmileyEH]

My usual rule is bet 2/3rds pot regardless of the board against one player and check if unimproved against two players. With position I will often bet against two though. More than two opponents its pointless to try and steal - you will be called >>50% of the time.

-SmileyEH