And carry a slingshot for backup!
Peen the bulges back down, IF there is enough room underneath, make a short tool steel strip for each side and SuperGlue them on (just to locate them). Have a good welder TIG them on at each end, which wont be directly over the thin spot so you wont get burn through.
I used to have a magician for a welder, big stuff to small. For jobs like this he would use really small tips on a gas torch and a magnifier. He, sadly, is gone now, like so many of our 'old school' craftsmen.