A leak of this severity should have left a wet oily mess in its wake. You don't need UV dye to spot that. Inspect the condenser coils, look at the compressor, the area around the service valves, the line set, etc. If there's no door that allows viewing/cleaning the evaporator coil then cut a hole in the ductwork so you can see into there (beware of the evaporator's tubing just behind the ductwork; you don't want to damage those!). Line sets can leak too unfortunately, especially if there's a joint somewhere, and if it crosses finished spaces then it's pretty hard check for this.
It's really unfortunate that service call cost so much -- before spending that much again I'd think really hard about system replacement. If you find the right contractor the next $1100 might be 25% of a new system installed.
Regarding shop air: it will contain moisture, could contain oil carry-over from the compressor, could contain debris degrading from the interior wall of a hose, could contain dust or whatever happened to be collected in the blow nozzle or other hardware/fittings you might use for connecting to the A/C system.
Then again, your leading alternative to fixing the leak is buying a new system. So if in the effort to find the leak you happen to do something that shortens its life a little or a lot, "nothing of value was lost." It's an opportunity to learn something anyway, and that might or might not have personal value to you.
I think you're correct that adding dye without refrigerant will be unproductive because the system cannot be run to circulate the dye.
One thing to be aware of is that normally an A/C system is charged with the refrigerant bottle inverted so that both oil and refrigerant are dispensed into the system. Too much re-charging can leave you with a system that doesn't work as well as it should because of the excess of oil (because the oil consumes volume that ought to be occupied with refrigerant gas/liquid). It's a bit of work to open a system sufficiently to remove some of that oil. I don't have any hard numbers to share, but I can say that most of the time when I replace a system I cut the tubing open and nothing drips, but I've had a few "we've had some leaks" systems that, when cut open, drained out a few ounces of oil and I had to work quickly to mop up the mess.
You could ask the service person to put dye into the system and recharge it with reclaimed R-410A. This is the stuff they capture out of other systems they're upgrading. Normally they would turn it in for recycling and receive some credit for it, but the value is small compared with the cost of new refrigerant. There is a risk that the reclaimed refrigerant may be contaminated and, like testing with shop air, this means a risk of damaging your system. But perhaps it's less expensive and worth the risk in your situation.