I think you are being really selfish about this.
You know and I know that Delaware is an "Inspection before renewal" state. You know and I know also that Delaware has no inspection stickers as their validation tags on the license plate technically serve as that. However, you are beating a dead horse here. Uber wants a valid windshield sticker form PA, period. It sucks. Its a stupid rule. I don't agree with it. But this is what they want. So if you want to keep driving for Uber, stop complaining anf put on your big boy pants and make some phone calls in PA for a safety only inspection near the border. Lots of places in Delco will do this for you. Costs $20 and its done.
"Oh its too far" blah blah blah. I feel your pain but you will not win. This is Uber. They do what they want. I suggest if this really bothers you to still get the inspection done then keep trying to contact Uber to get the rules changed.
On another note, I d0 know when you inspect your car in Delaware at one of their inspection stations, they give you a paper with the results. Why didn't you send that to Uber? Either way, do what you need to do to make Uber happy. They aren't going to stop everything and investigate if you are right, just for you.