The “browse update template” is linked to a browse, so it should be getting the record for whichever browse it is linked to.
The big question is, how will your procedure know which browse it is you want to call the update procedure for?
Once you can figure that out, it is fairly trivial to Access:Filename.Fetch(PrimaryKey) the correct record and call the update procedure, which will then update that record. Have a search in your source code for where the update procedure is called. There is an embed just before that where you can call this sort of code.
I always prefer to update the original file, rather than the alias - might not make much difference?
After returning from the update procedure you can also call whatever code is necessary to update the display of the appropriate browse.