Cleaning a jacket with an iron-on design

Factual Questions
1

As I posted in the current MMP, I found the jacket I was given as part of the Space Shuttle Support Team. I appreciate the freebie, but it’s actually pretty cheap. The design on the back is ironed on, and after 35 years there are a couple of places where it flaked off. Clothing tends to get dirty just sitting in a box, and I’d like to clean my jacket. I worry that dry cleaning will remove the design entirely. I could give it a cold wash with detergent and hang it up to dry.

So how do you clean an article with an ironed on design?

2

If it’s just dusty, I think I’d just wipe it gently with a damp cloth, something that’s not going to leave any little fuzzies behind. Maybe use a very mild solution of something like Murphy’s Oil Soap.