Doing it without pulling it all apart is going to take a ton of time and die grinder work. Like Fast V said, you need to get ALL the old globbly goop completely removed before rewelding. Pay a GOOD welder to do the welding as it will be a challenge to get proper weld penetration and strength in an already contaminated substrate.
As and addendum, you might even add gussets on all those rebuilt joints just for added strength.
When you get upside down, EVERY joint in a roll cage may or may not let you live through it. You build a roll cage to work only ONE time, the time you really need it and it better be strong !
I've seen race cars that passed tech, that I wouldn't even SIT in with the engine OFF, they are so badly welded. Scary, scary stuff, Maynard !