What about applying a rust converter - it works well on cars and farming implements etc so should work equally well on a radiator to remove rust before painting.
There is a product called Jenolite rust converter which you can buy off eBay or from Rejel - and think on Amazon too.
www.xxxx
I realise a few months have passed since post was shared, so wandering if you have found the best solution to combating the rust spots or if the car is still posing a challenge. If you are still working on your car here is my contribution to consider - Many classic car enthusiasts use a product...