I would just have the area affected by the root fixed. You could have somebody come out and cut the asphalt/concrete section apart from the "good" part, and then have the backhoe remove the smaller section, and the root. You could frame this up after they have taken it out (to save some money) and get some concrete poured to replace what was taken out. You may blacktop over this for uniformity, or just leave as is. You'll always have to take care of the blacktop...seal cracks and topcoat it. Curious, why was the concrete covered with blacktop in the first place?

Good luck!

I just found out there is a concrete grinding machine which can "sand" away the top of the raised part of the driveway, instead of digging up and replacing it. You can rent these at tool rental shops, but you also pay for the basket, which I guess is the part which actually grinds away the concrete, and becomes dull.

This machine is about the size of a mower, and you walk behind it. I saw in our street there were high sections of the sidewalk ground away. Then later I saw you can rent this machine to do just that: grind high sections of a sidewalk down.

If I were to guess you could rent one of these for $50-70 per day, or less for half a day.