Chromo used in bikes isn't normally hardened, so you won't affect it below the other welds or joints by welding in a bracket. You can pull alloy out, but that's really beyond the level of detail here.

Welding any tube can affect it structurally, so you probably wouldn't want to add a bracket in the most highly stressed part of the bike unless it was sufficiently strong. I mix in mild steel with Chromo in applications where stiffness is the main issue since it pretty much is the same stiffness. Drops would be an example.

Mig will work but you have to have the correct settings right out of the gate, and you have the cold start issue. A lot of chromo is migged including many stock car frames, but they had plenty of time to develop their proceedures on scrap. You are better off using a Chromo based wire, I had enough trouble getting that for TIG. You can also use a stainless all purpose. If it was up to me I would see if I could pick up some other old bikes in the trash and do a few coupons, just because of the settings.

Remember that you want loads tangent to the tubing since radial loads can crush thin tubing. In other words if your loads are runing up and down from 12 o'clock to 6, then you are better off (or at least should consider) two small tabs attached at 3 and 9 rather than on the radial axis. The tubing can be too fine to take a direct hit. Custom choppers and such can clean up the look with radial brackets but they often increase the wall thickness as much as four fold.