The front camera isn't actually part of the Face ID system; rather it relies on the infrared (IR) camera reading the dots that are generated by the dot matrix projector. The problem is that in general you can only replace the front camera as part of the front camera assembly, which includes the camera, the dot projector and the IR camera.
The guys over at a company called REWA discovered that you could actually replace the front camera as long as you kept the rest of the original parts on the front camera assembly, at least on an 11 Pro. While I can't guarantee it will work the same on a 13, I believe it is highly likely that it will. The thing is, it's easy enough to test. If you buy the camera by itself or as part of another full assembly that you'll disassemble to get the camera. You can go either way; the assembly is easy enough to get on places like eBay, Amazon and AliExpress among others, or you can buy just the bare camera on AliExpress like this one.
Oneten Front Single Camera Face Proximity Sensor Flex Cable For Iphone 13 12 11 Pro Mini X Xr Xs Max Mobile Phone Repair Parts - Mobile Phone Flex Cables - AliExpress
Either way, you'll want to verify it before you commit. That's simple enough; all you have to do is disconnect the existing front camera and plug in the replacement instead. Try out your phone and make sure Face ID and the front camera are all still working, then you can go ahead and do the replacment.
I have to warn you here that you'll be blazing a trail here, as there aren't any guides for what you want to do. The best I can do to help out is to start out with a guide for replacing the front camera assembly on your iPhone 13; sorry to say that iFixit doesn't have one, but there's a Spanish site called Nadie Me Llama Gallina, or Nobody Calls Me Chicken, lol, that does.
Manuals / Apple iPhone 13 / Front cameras
Once you have the camera out, you'll want to do something comparable to what REWA did on the 11 Pro; e.g., cutting open the camera housing. Take a look at their guide starting on Step 31. As I said, you'll have to figure out the exact thing you'll need to do on the 13 for yourself, as nobody else has done that particular repair that I'm aware of.
iPhone 11 Pro Front Camera Replacement Without Losing Face ID - iFixit Repair Guide
That should give you everything you need. When I was researching this same sort of fix for my grandson's iPhone X, I found a vendor on AliExpress that was selling the actual front camera assembly metal bracket by itself. A quick perusal didn't come up with any for the 13, but if you give it a good search you may find something that my short check didn't.
Finally, if you feel ambitious you can help out the rest of the users on this site by taking pictures as you go and putting together your own guide to this particular repair. Just go to the Repair Guides page and click the blue "Create a Guide" button in the upper right corner of the page. It's pretty easy to do and you can build off other's work for the steps leading up to the part you're explaining. You can always ask for help if you get stuck or have questions; just head on over to meta.ifixit.com and ask!
Good luck; let us know what happens!
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