Sounds like the paint is soft and if this is the case, get some BLACKFIRE Polish and skip the paint cleaner.
I type this all the time, here goes again,
A paint cleaner is an either/or product.
You either use a paint cleaner or a polish but you don't use or need both as this is redundant. Here's the deal. A high quality polish will do EVERYTHING a paint cleaner will do and more as it will actually remove defects. A paint cleaner only cleans the surface. It requires just as much time and work to apply both so might as well apply the polish and actually get better results for your time, money and energy.
FWIW
I never use paint cleaners. I'll either use a true polish or a one-step cleaner/wax.
If this paint ends up being finicky, you might like the new Pinnacle Jewelling Wax.