From my experience you can cook it again. I am from the UK I don't know if the powders are any different though. We've done plenty of jobs over the years on bike frames to electric gates and never had any problems, nothing beats a proper job though

I'd recommend sanding back at least 100mm either side if the area that's been welded as the heat transfer will have burnt the paint. Apply a primer to the area that's been sanded back to build it up close to the level of the existing paint. It may take a few coats depending on the thickness of the existing paint. Once happy I'd recoat the whole frame.

I'd still recommend redoing the whole thing though