Can I do a CAT III ILS approach if minimums are above CAT I?
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!