If you want to clean up the appearance of the welds, a material you might consider is the German made Devcon Stainless Steel / Epoxy putty. This stuff is pretty commonly used for non-load bearing repairs in my line of work. It's pretty durable and takes a reasonable surface finish. It has about one quarter the density of steel or brazing filler, so for a given size of fillet this method will not add as much weight (if you care about such things). It isn't cheap.

If you primarily want brazing practice, that's a different issue.