In that case: blow up the balloon, melt rock outside balloon. You could make "cement" from grinding local material plus some binder, that way you bring in less mass. Radiation shielding is also a nice use for asteroid's mass. I'm just saying that blowing up a balloon is easier than extracting several cubic kilometers of rock from inside of asteroid. You can start from that premise and try to find some solutions.
> Heating and melting foam is harder than solid rock or metal
You can grind it, then blow it with very hot gas onto your baloon, making a nice melted coat of rocks. This technique is already used to repair metal rotor shafts, it deposits machinable metal: https://www.youtube.com/watch?v=NAeBpF84Q9M
Humans need room–temperature nitrogen, not nitrogen plasma. You would rapidly make the atmosphere inside your bubble unlivable. There are few easy solutions here.
Then you wait a moment until it cools down. Of course there are few easy solutions, space is very hard environment mercilessly waiting for you to make an error. You can find problems in every step. It doesn't mean you should give up before you start.
> Heating and melting foam is harder than solid rock or metal
You can grind it, then blow it with very hot gas onto your baloon, making a nice melted coat of rocks. This technique is already used to repair metal rotor shafts, it deposits machinable metal: https://www.youtube.com/watch?v=NAeBpF84Q9M