Just to update everyone, I tackled the wheels again tonight. I first tried spraying on a diluted ONR and wiping it down with the ONR solution. No success, it did absolutely nothing to help remove the white residue.
Next I tried spraying some of the wheel cleaner on a microfiber cloth and wiped an area where the staining was. No success with doing that either. It's like the residue was baked into the coating/clear coat.
Lastly I resorted to trying Meguairs Ultimate polish. Fortunately, this did the trick. I rubbed the polish in by hand with a microfiber applicator and used a ton of elbow grease. After wiping away the hazy polish that I just worked in, I found that the white staining was gone and the finish looked nice again. I was able to get like 90% of the staining gone. It was tough to polish the areas that are tucked away and hard to reach, but for now I am satisfied, until weather gets warmer and I have an easier time doing a deeper clean.
However, on closer inspection I saw that my first pass of cleaning the wheels (when i first used the wheel cleaner) wasn't really great because I still saw caked on brake dust at the inner lip of the barrel, brake dust in the little nooks and crannies, and in the lug nut holes. Some of the brake dust is so crusty and I can't even remove it by scratching it off with a finger nail. How do people actually remove really old, built up brake dust like this? I'd like to ceramic coat my wheels again after the polishing but I have no idea how to remove the heavy brake dust and get the wheels super clean to the point that I can apply the ceramic coating. Thoughts?