First time I"ve seen this.
If you push, Billy Big Stack will fold pretty much anything, other than AA.
If you call, he"s going to call as well, so you get more chips in the pot. If the flop comes Axx, then you check it down. If it doesn"t have an ace, you make a bet on the flop to increase the chips you want to accumulate.
Checking it down regardless isn"t a bad option either, especially if you"re treating it as a satellite first and foremost.