Can I do a CAT III ILS approach if minimums are above CAT I? I was going to an airport with weather at minimum for CAT I approach. Not wanting to know whether or not I could make it I thought about asking ATC if I could do a CAT III approach even though the weather was still above minimum for CAT I. Eventually the weather got better but my question is: Could I have requested from ATC and be granted a clearance for a CAT III approach (abiding by the CAT III weather mins) even thought the current weather was above CAT I minimums?
Yes, you can request it. We (ATC) may or may not grant it. It can be quite complicated for us to establish CAT II/III procedures, including having to switch over to backup power, turning on stopbars and other protective measures, limiting aircraft and vehicle movements on the maneuvering area, closing certain taxiways etc.

Doing a CAT III approach is not just looking at a different chart, it can involve a lot of complicated procedures for the airport!