I did find a guide on a Spanish site; it's got an option for the English language, but it didn't seem to work well until I allowed Chrome to autotranslate, and after that it seemed to be usable.
Manuals / Apple iPhone 13 / Front cameras
I can't be sure about the iPhone 13, but on the 11 Pro at least, you can replace just the camera without losing Face ID. It involves taking a Dremel to the metal housing and digging out the camera, leaving the dot projector and IR camera in place. You can get the general idea from viewing the iPhone 11 Pro guide that I helped out on.
iPhone 11 Pro Front Camera Replacement Without Losing Face ID - iFixit Repair Guide
I was looking at the same situation on an iPhone X and was able to find a site that sold just the camera alone without the rest of the front camera assembly, along with a replacement metal bracket that could be used if you decide to just destroy the old one in the process of disassembling it. Turned out to be the IR camera that was causing the problems so I didn't end up doing the repair, but same logic applies; use your specific guide to get the camera out, then use the same technique to remove the camera from the assembly.
Good luck; let us know how it goes! Take pictures; it would be great if you authored a new guide to that replacement for all the other people who are going to run into the same problem. It's actually pretty easy to do, and there's plenty of people, myself included, who will be happy to help you with it.