Would they be able to look up my information to verify? I have the emails with my keys but they’re also available in my account on my.uniden.com
Your "keys", as well as the 'receipt', showing that you purchased them, are 'secured' in your "My Uniden" account. I doubt that any of the techs, at the repair facility, would have access to that information, unless you furnished them your login credentials, which is not a good idea. (Neither is that a 'good idea' to share logins for any other online accounts registered to you.)
Your better option is to copy the receipt information, as well as include the ESN that was valid when the scanner was originally updated. Point out, in a message to Uniden, that your original 'physical' serial number, is based on what is labeled on the scanner itself. But the ESN, on the replacement logic board, is different. Uniden should have 'flashed" your original ESN into the replacement board at the time of repair. I would suggest contacting Uniden, with a message including your original receipts, and note that the ESN of the replaced board differs from the one that was valid when your keys were purchased.
Hopefully, whoever you are dealing with at Uniden's repair group can get the keys re-issued with the new, modified, ESN from the replacement logic board. I do wonder, however, that if the ESN on the replacement board is significantly different from the physical key (portions of the original key are part of both the physical serial, as well as the ESN, would that mismatch pose a problem in getting valid replacement upgrades to validate and allow installation.
