3 betting OOP should be 4.5x-5x. We're really not looking to get called with any of our holdings in theory (even AA.) Playing OOP pots in general is a nightmare because we under realize our equity don't get the information that the IP player gets.
With that being said, AK is just too strong to be flatting. I wouldn't say to 3 bet 100%, but I'm going to be at least 99%, and it should be go to 3 bet.
For the hand I would go $45 pre. With your sizing, he's presumably calling a lot of his range that he's not 4 betting, and filtering out the bottom. With the given information I put villain on a range of 22-JJ, suited connectors from 65s to T9s, suited gappers of 86s to J9s, AJo-AQo, KJo-KQo, all suited broadways except AKs, and then QQ at 50% frequency, AKo at 75% frequency, and AKs at 50% frequency.
With the given range, I'm saying hes calling a flop bet with all middle pair and better, all of his FDs, all of his gutshots, and all of his OESD. With the help of Flopzilla, we can see that villain is continuing on the flop 78.92% of the time. With that being said, even a 1/3 bet otf, which needs to get through 25% of the time, would not be profitable here. I'm x'ing to x-f, unless he uses really small sizing, because the only turn card that I feel completely comfortable with is a Q.