It is a shitty spot when that happens. I have deleted 3 responses so far after typing them. I think I like betting here for balance and hand defense. Maybe a delay cbet on the turn to prevent being floated, and generally cause confusion. Many players might put you on a weak pair with this pattern and give you some value, and theres some chance of scaring off stronger weak aces with this line too. All depends on the opponent type of course.
Overall I think its a rough spot to determine optimal action, which means that assuming you don't royally screw it up any reasonable course of action is probably fairly close in EV. Best thing to do is probably just not play weak enough aces that you feel you have to worry and try to get to showdown cheap, but still defending your very vulnerable hand.
High fold to cbet often means either total nit, who you have to be worried about having a stronger ace, but might pay you off with kk qq and the like, or a total preflop station. So you've got a lot of things that are seeing flops and then ditching when they miss. These types will either let you know when you have them beat, or be superpassive, so your defensive bet will probably be the only one you have to mess with to see the showdown, or be told plainly you are beaten. But against the lose guy you'll wind up letting him draw to 2 pair and calling his river bet thinking you have him beat and he's just trying to steal it from you. You may as well be the aggressor if you're putting in the one bet anyway. Then you still have all the SD equity you had in either case, plus fold equity.
Incase you couldn't tell I decided to stop deleting things. I think the stream of consciousness thing is a big help in poker-think. Rambling conversation that went like that were a big part of how my game grew, and many of the big pros have said the same, so screw it. Enjoy my novella.