I think you probably have to go with it, but I don't think you really have the equity that those poker stove calculations suggest. The probability of him having AJ-AK here surely has to be discounted - I think the 3-bet means you have to weight his range towards AT/TT/55. As does the fact that he might well have reraised preflop with AK. And the case ace also makes up a decent chunk of the shortstack's range.
If you discount the chance of him having AJ/AQ by 50%, and include only one combo of AK in his range, then I think you still get about 35% equity vs a range of AT+/55/TT, which is approximately what you need to break even.
If you take this view though then I'm not sure there's any point jamming the turn - you might be giving the weaker parts of his range reason to fold, even getting 3 to 1. AJ's going to struggle to find anything it can beat if you 4-bet it here. I think I prefer to call turn and then call or value bet river.
good post.