Painting won't cause it to explode! I can't even imagine what was going on in the mind of whoever originated that idea.

On the other hand, paint won't fix spots weakened by rust -- I'd recommend explaining that in writing to your customer and getting them to sign off on it. If the rust is nearly through the shell, it's likely (if you have my luck) to start leaking the week after you get it painted and all put back together. Customers in that situation tend to expect that you -- the service person -- have some obligation since "you just worked on it!". Explaining it in writing -- ahead of time -- and getting them to initial that can save your . . . 'rear area'.

As far as what paint -- any light color that will adhere well to metal is fine. To paint stainless, you'll probably need to go over it with a Scotch pad and detergent or TSP first, and then use a primer. Otherwise, there's a good chance it will start peeling off rather quickly.
Epoxy pool paint would be very durable, but is not usually UV protected and will 'chalk' over time.