Ok, I'll just clarify those suggestions for the sake of liability and because I'd love to see it work.
The wrinkles and waves indicate that the metal was stretched slightly during the hole cutting and recessing operations, that is to be expected since this was a pan not a sink when it was stamped and formed. Also the hole is dead center of the largest plane. I'm guessing you used a hole cutter that is a punch and die knock out style and then a similar system to recess..? Stainless can be a bear to cut with saws. So you just want to slightly shrink the area to make it pull taught again. This is the method you may have seen used to repair hail damage to cars in years gone by. Those dents were heated up, usually with a torch or industrial heat gun, but not enough to melt the paint, then a small hunk of dry ice would be placed in the dent and it would shrink the metal and super cool that area, and since it was a focused spot operation, the area around the dent was still stable and would only allow it to shrink back to it's original shape or memory. So you don't need it to be white hot but it does need to be made more "plastic" and free up the molecules, then slam on the brakes with the dry ice and lock in the memory. I just thought that a BBQ could heat evenly and you could do it out side. There is not a lot of mass in dry ice, but it is super cold and can make the shrink happen instantly while the molecules are moving free-er. If you put a fender washer or two bolted through the recess, that should stabilize that area and allow the shrink to happen around it. It doesn't take much stretch to make a wavy area since you are effectively trying to make a 6.25 x 10.375 piece fit into a 6.0 x 10.00 hole for example. If it can't go out, it will go up and/or down. I'm sure Icon has got a handle on the concept here.
Disclaimer; Never, ever do anything. Ever. Something could go wrong. McDave is not responsible for any damages to person or property. Fun is an unintended by-product. Your results may vary.
Yeah Hank, still got all my stuff, it's just worn out. Especially right under my hat and from there down.

McDave