Sounds entirely wise to me. In the old days, people cleaned huge telescope lenses with wine and a chamois. You might try the Kodak lens-cleaning tissues [I've liked them over the years, wrapped over a cotton swap for small surfaces]. Also, I do not fear using Windex[R] [glass cleaner with various useful additives]. Modern coatings are very resistant to these things. The wetting agents and streak-reducers in Windex[R] seem to work well.
I might add: laundering the microfiber cloth may be wise.
Unless the haze is severe, I wouldn't expect noticeable effects in printing. I say that because I have a mid-1930's Rolleicord with non-CLA'd, uncoated lenses, and it makes super photos. The lens is decidedly dirty, but has no fungus.