PF try going smaller like $.17. Then the standard is to c-bet a flop like this 1/3 pot. Then I would probably just check this turn. I think you're over thinking it. Triple barrels are almost pointless at this level.
I'm gonna assume they're a weaker player (coming from me haha) seeing as they limp called pre.
I think once they call flop and turn, the runout is dry enough that they're never folding river. They're basically saying they have a jack, so it comes down to will they fold it thinking you have AJ+, and the answer is... idk. I don't think so
Typically they're not gonna fold top pair
Also I think in theory AK has enough SDV that you'd rather check it behind and bet something like KQ or K9, which also blocks the straight. IDK, I just find that the type of players who limp call, then call flop are almost always very top pair heavy, and will call you all the way down unless you fire 3 barrels and the board gets 'scary'
I'm expecting to get called quite often by Jx here, maybe some middle pair that rivered 2 pair
IDK about flop sizing, I wouldn't be surprised to find out this is a spot where the solver has a mixed strategy incorporating some 1/4 and some 3/4, maybe even some overbets depending on what you think their limp calling range is.
I agree with doodoo, AK is your best non-made hand to try to check all the way down to the river, use your KQ/QT type hands as the low equity bluffs if you want. Even if you had the flush draw otf, I think you should check it.