You could try, but it would be better to replace the frame. If there is rust in the rail, it is very likely more extensive than you think. Another spot for rust in in the kickup over the rear axel. Be sure to check the entire frame for rust before thinking it'll "just" be confined to one place. A hammer, some small mirrors, a flash light and ice pick are your best friends.
There is a web site that shows exactly what you are talking about. A guy inherited a 4-door 65 Galaxie, found a little rust in the frame and replaced one side, found more rust - which lead to a full frame off restoration. If, a pretty big IF, the front and rear suspension sections are rust free, the side rails are straight dimensional steel. It is possible to replace the section.
Good rust free frames can still be found and purchased. The good thing about having a replacment frame, is you can detail it before you install it. Get everything right, then separate the body from the old frame, clean everything and put it back together. If you are lucky, you may not find any floor sheet metal damage...maybe. Then all you have to worry about is saving as many of the bolts as you can.
Remember...Rust never sleeps. If you don't get it all, it'll just resurface.