Ask Mr. Smarty Plants

First of all, removing that elm stump, if it involved a stump grinder, would almost inevitably hurt the live oaks, not a good idea. So, if you're willing to live with the stump, we think that's fine. You do need to keep your eye out for elm sprouts, as that tree is going to be fighting for its life, with seedlings popping up and suckers coming off the stump and roots. Keep cutting them back or yanking them out and they will finally give up. And don't put herbicide on any of the elm tree remains, either, as it could manage to get in the soil and damage the live oaks.

Now, in answer to your question. Yes, some new growth will come in on the formerly crowded side of the live oaks. It won't be a quick process, but at least it will get better. But don't, repeat, do not trim the live oak now. We are now into February and the Nitidulid beetles will be active from now through May. Oak Wilt, a dread scourge of oak trees, and especially live oaks, is caused by a fungus introduced into the trees by the beetles. Cuts or wounds in the tree may offer an opportunity to those beetles. It would be a shame to go to all that trouble to encourage the live oaks, only to have them succumb to oak wilt. The safest thing to do is take good care of the oaks, making sure they are getting plenty of water, and wait until a cold spell in January to do a little shaping up, taking some off the side that was able to grow. Don't take off too much at once, do it gradually, until you get to the point where the sides match up a little better. The tree needs the nutrition those leaves are providing, and if you cut off too much at once, it could be seriously stressed out. At this point, we don't feel fertilizer is an especially good idea. Those trees have been getting along just fine with the native soil around here for millions of years without any human intervention. It might be a good idea to mulch the roots with a good shredded hardwood mulch, to help hold in water and protect the roots and soil from the heat. 

And don't forget-watch out for little elm trees trying to regain their place!