I've had to clean this sort of thing off before and I've learned to work carefully.
I would *not* use that product on it, the plastic "chrome" can be mighty fragile. Try to use the mildest stuff that'll do the job, starting with something like AIO or a cleaner wax. I hesitate to recommend chrome polishes as some of them are awfully harsh, but a mild one of those is about as aggressive as I'd go.
The white spots aren't like the corrosion blooms you can get from a compromised "real" chrome plated surface because the "chrome" stuff is just applied over the plastic, it's not one of many layers of metal platings (copper, etc.) like you have under real chrome.