How to treat rust spots on a steel frame without stripping the paint completely?
I would lightly rub the offending area with steel wool. The idea is to not completely remove the rust but to dislodge any paint that is intact but not bonded to the metal. Apply a rust restorer or converter. Rustoleum is one brand but there are many others. Apply the converter to the prepped areas per the instructions. The chemical process converts the rust and leaves a coating on the bare steel(typically black). You can then cover the areas with either touch up paint or nail polish of a similar color. If you visit a Hobby Shop they have small bottles of enamel paint that you could custom blend to get a close match.