Typically, phosphoric acid is what's commonly used for cleaning bathrooms because it cuts through soap scum like a hot knife through butter, but it won't harm chrome.
If your allergies act up when you clean the bathroom, is it because you're allergic to mildew spores? If so, then you should be aware that it's possible to clean mildew off of silicone caulk, and it's also possible to clean mildewed grout.