Think flatting can't be too bad. When BB is reasonable he won't be squeezing too wide.
Don't like 3betting.

No it plays horrible oop postflop and getting squeezed is a disaster

If BB is halfway decent, he should be calling close to 100% after you call (or 3-bet). That means we are very likely going to the flop 3-way by flatting here.

Our hand plays okayish in this 3-way pot with the 2nd nut FD and 3 broadway/1 wheel card(s), but the lack of nuttiness kind of hurts. I still think calling here would be +EV (=better than open folding), but not by much. Also, what was the opening size? Just stating the obvious, we should be inclined to call more often against a small sizing - although that might get a bit complicated if BB starts to squeeze more often.

Now, the question is whether the EV of 3-betting can be better than that of flatting. This, of course, depends on many other things, but I think it's a decent option for roughly 2 reasons:

a) We can knock out BB quite often and get HU with MP; BB also can't 3-bet and put us in an awkward spot
b) We block A and K, which are great blockers to BB and MP's 4-betting range

And given that people tend to play poorly in a 3-bet pot, we might as well be able to increase the EV compared to when we flat. Once again though, this depends on many things and I won't try to compare those two here. Personally, I think I can play this hand more profitably against most opponents by 3-betting, although flatting wouldn't be terrible. With AKQ5 with Q-high suit though, I'd be more likely to 3-bet.

I'd never really considered 3 betting until midori mentioned it and why. Midori raises some valid points imo, knock out bb, A/K blocker, cleaning up our non nut suit. I don't think I'd like to 3 bet it every time because we will end up 3 betting too much but against some opponents who 4 bet rarely and fold to a lot of c-bets it could be a good idea.

Edit: I'd like to add that if MP (25% open) raised and we are in the bb I would call 100%, whether it would be going multiway or HU.