Who cares if the Democrats file suit? How many suits have been filed against Obama? Did it stop him? Slow him down? No. He assigned a lawyer (or a couple of lawyers) to handle the cases, and he moved on.
The same would be true of Cruz. He would have a couple of low-level attorneys handle the lawsuits (just like any nuisance suits filed against a Presidential candidate or President - and there are plenty of those, for every President), and move on.
Me thinks you missed the point of my comment, but that’s OK.