Currently, the problem is not fixed. The service department followed their process. They replaced the fuel pump and then the fuel sensor. Neither resolved the issue. This exhausted their fixes, so they opened a ticket with Audi. So far, Audi's solution is a "special" fuel pump from Germany. It has yet to arrive or be installed. Nobody is sure it will even resolve the issue, but they are all hopeful.

Keep in mind this is a brand new car that has 68 miles and was delivered with 25 miles. The car was in my possession for less than 24 hours when the check engine light went on. I had the car for 39 hours before it went back to the dealer. No parts on the car are broken, based on what I've been told. To me it looks like Audi is shipping out cars with a fuel pump that does not deliver the required amount of fuel (or fuel pressure?). If so, that's a design flaw and is inexcusable.

I did some internet research that turned up similar situations for the 2018 RS3 and a few other models: fault shows not enough fuel, pump replaced, sensor replaced, car is in the shop for weeks/months, nobody knows what is causing the issue.

Update from Jul 25, 2019: My guess is that the "special" fuel pump is overkill - delivering far more fuel than the car will ever need. However, it did not resolve the problem. It turns out that on the opposite side of the fuel tank from the fuel pump is a fuel line connection (I think they said it was a T-joint or something like that). That connection is supposed to be secured, but came from the factory unsecured. My understanding is that the securing clip is there, but the fuel line was not attached to it.

The high pressures from the fuel pump causes the fuel line to move around (think fire hose on full blast with nobody holding it). If the fuel line is not secured at that connection, the connection can come loose, resulting in low and erratic fuel pressure. The fix is pretty easy - put the fuel line in the securing clip/whatever so it won't move around at the connection. Of course, it took 3 weeks to figure it out and involved the top Tech at Audi North America.