But not the same mitt you use on the paint on the car?
I dissected what you wrote below to put it in a different order...
If your rims or wheels are factory painted, and Eagle One Wheel Cleaner stained them, then you might want to contact Eagle One and let them know, especially if the product in question states it's safe for factory painted rims.
If your rims are factory painted, then you would treat them like car paint and use the appropriate paint cleaner to "try" to remove the spots. Test any paint cleaner in a small area first and see how it's working and what the results look like.
I've been told the paint used to coat wheels is different than the paint used on car bodies, I point this out because I think some people think that if a car has factory painted wheels it's the same type of paint that matches the car but that's not the case.
Here's the deal...
If your wheels are painted and the spots or the stains are topical, that is the offending substance only stained the very outer or upper layer of paint, then it might be possible to abrade the paint with a light polish or chemically clean the paint with a non-abrasive paint cleaner and remove the spots or staining.
If the spots or stains have penetrated into the layer of paint to any depth then it might be more difficult to restore the finish.
Don't know if these wheels are a big deal to you or not, but if they are expensive and important then before you start applying other products you might want to first contact Eagle One.