First off I know this post is old, but i am a professional landscaper by trade, 20+years. If you cut out a circle of grass to replace when you are done with the pool for the summer. You will have to water that circle of grass twice a day, or it will die also. It needs soil to keep cool/Warm, and also to keep the roots from drying out, and keep nutrients to the roots. So doing this to keep one circle of grass alive would be more work than moving the pool everyday.
That being said, if you are going to plant any kind of ground cover as in suggested above, clover, barley, ivy, etc, make sure you make the diameter of the circle, square, rectangle or whatnot, already a foot or two larger than the pool all around.ground cover will spread quickly to find Sun and "IT WILL" take over your lawn fast. So my suggestion making the diameter larger is to put landscaping fabric down in the cut out area to keep the roots of the ground cover from spreading and also put some sort of edging to keep the roots of the ground cover at bay, and not able to creep into the law. 3rd when your grass grows back like you say, i guarantee most of it is weeds, and not actual graas, weeds.spread much faster than actual real grass.
Grass will spread but not as fast and seed cover. You could buy a bag of grass seed and throw some seed all in the area, scratch the dirt with a cultivator or a heavy duty metal rake, than seed, than take a little more and take the seed in. Also water every.day for.the first two weeks, twice a day for 20 minutes min, and water in early morning, or just at dusk.
I think your best bet is Astor turf, or that outdoor rig stuff they use in boats and outdoor patios sometimes. If you dont maintain it with fertilizers three times a season, seeding in fall, and spring, weed control, then i again guarantee your whole.lawn is weeds and not grass. And if that's the case who cares about the dirt spot. Paint it green with green spray paint.