4-betting and getting it in can't be too bad, but given how tight a lot of players are with 4-betting at the small stakes, I could see just calling. If you do that, you're just going to have to put him on a range of {QQ+,AK} for seeing the flop and proceed accordingly. Probably that will mean folding whenever an A or K flops, calling most bets if it doesn't, and then making a decision on the turn about whether you think he'd bet again with AK (probably, given assumptions we're already making about the player, I'd fold to most turn bets).

What does your 3-betting range look like in this spot (SB v BN)?

You don't have to have that info to make a decision. If you don't feel confident guessing what his 4b tendencies will be, then you should just construct a balanced 5b range. Given the width of your 3b range, QQ probably belongs in there for value.

Yeah, actually, I want to back up a bit, because I think the stated 3b range is too tight, especially against in a field where 4bets are not common. Honestly if you aren't going to get 4b much I'd probably just never flat SB v BN, but even a conservative 3b range that allows for some flatting should still be at least 10% of the deck, something like: AJo+,ATs+,A5s,KJo+,TT+,54s-T9s,J9s,T8s, and perhaps also the smallest pairs (22-55) that aren't good enough to flat. That's 12.5%. Personally I'd 3bet more like 17% and never flat.

If your opponent 4bets so tightly that you can't profitably shove QQ, then he's leaving a ton of money on the table and letting you realize a lot of equity. So although jamming QQ will be less good than calling vs a player who only 4bets {KK+,AK}, you're essentially already winning against that player by realizing so much equity with your 3betting range. The narrower your 3betting range, the less you exploit this mistake.

If you never 5b jam, or only jam AKs and AA or something, then you risk making the same mistake he is. That is, if he has an appropriately wide 4betting range, or anything close to it, then you let him realize a lot of equity with those hands that would fold to your shove.

Bet turn. As played, bet as much as you think he'll call with AK (don't be shy, this is probably a bigger number than you might think).