the guy mentioned that he is going to put some sort of solder on the tubes that appear to be leaking and than pull vacuum on the radiator to pull the solder in. He went on saying that the aluminum racing radiators have a tendency on doing this if they are not mounted properly and the radiator vibrates in its place. According to him there is more chance of it happening when cars have stiff suspension and are driven on the street.
I mentioned having it cut and welded like you described and he pointed out that it wouldn't be worth it as it would be too time consuming and costly. He said most likely he can fix it, if not no charge but I would need a new radiator.