I would raise just a little bigger the first time, because of the coldcaller in the middle. x3.5-4 instead of x3.
As played, that's practically a min-raise. I'd shove.
Experience
tells me this is often a middle pocket pair looking to flip vs AK.
It wouldn't be a flip vs QQ.
Occasionally this is a donk-play with AA/KK, but my money's on 66-99.
Edit: I'm a 6-max player, though. Full ring might be different.