So, I'm planning on building an Uzi that will be an SBR.

I'm using a receiver repair stub to build on.

Do I make my own serial number up for it?

Or....

If I end up welding the rear part of the receiver onto the stub, can I use that serial number?

Thanks for any help.

Yes, if there's no serial number (and maker's info) engraved... then you're the maker, so you engrave both (and choose an acceptable serial number).

You can adopt that existing serial number engraving, if their serial is made up of acceptable alphanumeric characters (which I think UZIs were).

See the guidance in the first section here: https://www.atf.gov/firearms/firearms-guides-importation-verification-firearms-ammunition-firearms-verification-overview

Then scroll down to ATF Ruling 2002-6 regarding adopting foreign serial numbers (or serials with foreign characters).