I think the stack sizes being what they are, shoving is the right play. Minraising actually looks a little too strong. Even if overpairs are unlikely to fold to it, it has the effect of making them wary and you might not necessarily get their stack on the turn or river, especially if an action-killing card falls like Figaroo and Minh mentioned.
If you go all in, BB is looking at calling another 70bb to win ~145bb. Big unpaired broadways are too likely to be c-betting OOP with just overs into two players on that kind of flop, so I think the majority of the time he indeed has the overpair and you're taking his stack. In addition to the sets, there are several combinations of pair+gutshot type hand in your range like J10 or J9 that he'd interpret as you trying to use your big stack fold equity.
Then CO will be looking at even better pot odds
. The best hand he can reasonably turn up is QJs, in which case he has 8 outs and has to call. With TPTK hands like AT/KT, decent players are unlikely to lose any more money in this hand. On blank turns they would likely deduce their hand is beat and fold if BB fires again, or check to you then fold so you're usually not going to get more money anyway. Therefore, springing the trap one street later really just risks scare cards slowing down BB, which we don't want to happen.