Second, if they've fixated on the separate door that can be locked, float the idea of a formal written agreement drawn up by a lawyer. If you are going to be locked out of 40% of your apartment, then I think it's only fair that they should pay 40% of the ongoing rates, taxes, utility bills, service charges for common areas in the building/lifts etc. And a payment to you as the live-in caretaker/cleaner. This needs to be in writing.

If they try the old standard crap that 'family doesn't need lawyers', well, I work in the legal profession in a branch that deals a fair bit with the aftermath of that statement. Very profitable for us but a source of immense family trouble/drama for the participants. If it's family, then you really really need a written agreement drawn up by a lawyer. Trust me. And they've already red flagged that their expectations may differ profoundly from yours.

Third, this is going to be principally YOUR lifespace (and Chook's). To be shared as and when required. If they want to make a contribution to make this possible, then what they get should be proportional to what they contribute. At the very least.

Fourth, they aren't being tactful about this. Or maybe the tact got lost when faced with the prospect of a private apartment in the city, with someone else doing most of the work needed to keep that apartment available to them on a moment's notice. And who will be paying the costs of keeping that apartment available? Perhaps a sit down session to clarify EXACTLY what they want and what their deal-breakers are. Because their expectations may well have changed once they got a better look at what might be available for the price point.