Alternatively the BBs small stack size could allow you to do something a little different. If you 4bet small (to no more than $6) you allow the BB to put in the rest of his $$ and you can still 6bet shove? Say you make it $6 and the BTN just calls (of course if he pushes that's great too) then the BB goes AI you can shove.
While we don't have any solid reads, I think the chances of the big blind pushing all in before the flop of his own volition have gotta be pretty close to zero. He started the hand with 38BB and he's cold called a 10BB bet already - that spells loose-passive drooler to me.
Even if he were likely to shove though, we've gotta be very careful with the bet sizing if you wanted to try that plan - it definitely couldn't go above $6. Making it $6 is a $3.50 raise. The big blind only has $3.55 behind after that, and we need his shove to be
at least $3.50 on top so the betting action gets re-opened. If we raise to $6.50, for example, he'll be shoving for less than a minimum raise and we'll only have the option to call.
I prefer the $10 (or more, but that's my favourite number here) play for two reasons. One is that it definitely commits the big blind's chips - I doubt he's folding anything before the flop, but if we bet $6 and he just calls, there's a chance he'll fold on the flop if he doesn't connect and we won't have gotten his last $3.55.
The other reason is we're giving better odds to drawing hands - it'd be $3.50 for the button to call in a $11.25 pot, and the big blind gets $3.50 into a $14.75 pot if the button calls. Even if that's still not quite the correct price for the big blind to call, they'd be making less of a mistake by doing so. We want them to make a bigger mistake.
Wow... that was long
. Cliff notes version: I'd only attempt that (raise to $6 with the intention of six-bet shoving after the big blind pushes) if I knew 100% that the big blind was definitely pushing. And I don't see how we can know that.
As for folding AK in this spot... what, for $1.50 more in a $6.25 pot? I'd certainly be at least calling.
Besides it's the big blind that's truly OOP, not us, and I think some of the button's positional advantage is negated. The most likely scenario if we flat call with AK here is the big blind checks all the flops he doesn't connect with, we make whatever action we feel is appropriate and then the button is left with not one but two opponents to act after him if he bets / raises.