GF,
Technically, an circling approach ends at the MDA, so one cannot use the ILS, but must circle from the LOC approach.
What's your reference for this? Our country rules certainly have no such requirement, and practically, it doesn't make sense. Why does it matter how you get to the circling area (it may even be via a radar vector)? An ILS to a circle would be safer than a LLZ as you have vertical guidance down to the MDA.
Actually, a circling approach
starts at the MDA (or DA). You cannot circle in IMC. A "circling approach" is in reality a visual manoeuvre
after you have completed the instrument approach.