Flight Systems has been doing remanufacturing of Chrysler's computers for decades and they also did not just the SRT-4's "Stage" kit computers (as well as packaging the entire kit together and shipping to the ordering dealers) but also were responsible for the older Mopar Performance computers in the old Shelby and turbo Dodge models of the '80s and '90s. If all you need is the VIN flashed on a "Stage" computer they should still be able to hook you up.
If you need it in a hurry that $110 your dealer quoted isn't too terrible and they're likely quoting you for an hour labor. Just to flash the VIN shouldn't take more than 5-10 minutes but if that quote includes installing the computer you might ask them if you drive the car there and plug the other computer in while you're parked if it would be any cheaper. Then take it back out, drive home, and permanently install it along with the sensors, injectors, etc.
Eric H. - '04 Neon SRT-4 (eBlue/S2) - '04 SRT-4 (blk/stock & now sold)
'92 Dodge Daytona IROC R/T --
'91 Dodge Spirit R/T
plus a couple other turbo Dodges, a Cobalt, a couple Focus, big S/C wagon, and some other stuff...