Well, both shoving and calling are crap if he is tarping with QQ/KK/AA.
Based on my experience with this player I think he would tarp with KK and AA a lot of the time and tarp with QQ some of the time. Its difficult to imagine him flatting with AK, I think he would shove that to isolate. I also think he would shove JJ most of the time.
So that leaves the mid pairs and the Axs that he doesn't want to fold to the short stack's shove but will likely fold if I shove (after a five-minute tank). TT, 99, 88, 77 some of the time. AQs, AQo, AJs some of the time. So there is a lot of combos I can fold out if I shove, and then I'll be getting a huge price to take on the shorty.
Key thing here is the short stack's range. I don't think he is thinking about how short and desperate he is, I think he has to have a good hand. I think the big blind knows that the short stack has to have a hand. Therefore both ranges tighten up. If the short stack had a wider range I would happily ship the

to isolate.
How tight would the short stack's range have to be to make my shove unprofitable?