How do I avoid the overbet shove here?

[HelixTrix]

The hand he showed up with after I shoved - 87o. Yeuch. Next time I'll shove pre against that guy. Anyway, having analysed it I think I can show that check-raising is more profitable than shoving.

Fairly optimal raising range from 3 off btn with M = 6:

AT+, KJ+, KTs, QJ, QTs, JTs, 77+

This entire range flops a made hand or a draw so it's probably reasonable not to attempt to refine it based on his check as he presumably mixes in some checks with both types of hand.

Assuming he folds (AQ, AT, KQ, KTs, QTs) to a shove

and allowing for hero calls with (AK, 77, 99, TT) as well as (AJ, KJ, QJ, JTs, AA-JJ, 88)

that's 43 folds and 85 calls, giving me ev of:

43/128 + 85/128(0.28*3) – 85/128(0.72*2) = -0.06 pots

Now let’s assume if I check he bets AK half the time, bets all the other hands he would have called a shove with, and bets QTs as a bluff. We’ll assume his bet size is 2/3 pot. He folds (AK, 77, 99, TT, QTs) to a check-raise, and calls the rest. The rest of his range he checks back with the intention of showing down and often calling a river bet.

So he bets 82 hands, of which 27 fold to a cr and 45 call. My ev is:

46/128(0.4) + 82/128*27/82*5/3 + 82/128*45/82(0.26*3) - 82/128*45/82 (0.74*2) = 0.25 pots