Correct me if I'm wrong but I would assume that if you're cleared for the ILS, you are implicitly cleared to enter the ATZ, even though you fly the ILS for cloud break purposes and not with the aim of an eventual landing.
There's just too many "what if's" if you get a clearance like "Cleared for the ILS approach, report VMC, do not enter the ATZ". What happens if you do not become visual before entering the ATZ? You are most likely below the MSA now so you can't simply circumnavigate the ATZ. Something that would be tricky with a moving map GPS, and almost impossible without a GPS anyway. Obviously you can try to climb above the ATZ but practically speaking that would mean that the go-around decision needs to be made at 1200' or so (*) - which makes the whole cloud-break procedure pretty pointless.
(*) Assuming a standard 3 degree ILS and a 3-nm, 1500' ATZ. If you want to fly the ILS to the furthest extent possible, you need to go-around at about 4nm/1200' at the latest, in order to be at or above 1500' at 3nm.