If you are worried about water collecting in the bottom of the "pocket", you could try drilling a small drainage hole at the bottom of it. But really, why not just go with the flow and have the tree taken down. If the current damage heals, you are still vulnerable in any future storm. What would you rather have five years from now, a struggling, mis-shaped Bradford pear waiting for the next storm, that's still lifting the sidewalk, or an attractive replacement tree just reaching it's stride?
I'm as sentimental about plants, especially trees, as the next person, but sometimes you have to make the leap of faith and remove a specimen that's not working out. I lost a large plum tree in a thunderstorm years ago, and was heartbroken, but the tree I replaced it with (a zelkova) has worked out much better in that spot, and the experience has taught me to try and be objective.
Cheers, Sue