Are you obliged to keep all grass within the airport perimeters in a condition, which allows heavy vehicles to use it without getting stuck?
Hmmm, that's an interesting question! I thought I knew the answer, but I don't. So I had a look at UK CAA's CAP 168 first, and Annexe 14 (incl Amendment 11) second, and failed to find any specific reference to unpaved surface maintenance and condition except where it is part of the manoeuvring area, ie runway, taxiway, apron etc.
That's not to say there isn't a reference; Annexe 14, and its Amendments, are very long! I only looked in the obvious places.
There is of course the RFFS requirement to get to any part of the airfield within a stated time. If the RFFS has to use paved areas only, either occasionally or every time, that requirement still applies.