I wondered about that too. As long as the cell or battery used is of sufficiently low impedance to maintain the line loop. Certainly worth a try! Could have a battery box with a capacitor across to reduce the speech impedance path, although a bit convoluted. You'd need to measure the current first as a 'base-line', as the battery would need to drive enough extra current round the loop so that sufficient voltage was dropped across the mic, and the state of the rest of the line resistance (and exchange equipment) is unclear.
Evidently, the line in question is 'miles from anywhere'. I thought I was miles from anywhere but I still manage to draw 40mA round the loop. I don't know how you'd go about reporting a suspected H/R fault to Sky. Perhaps they'd say that 'as long as the line worked' - which, presumably it does on a normal (modern) telephone - there would be nothing they could do about it. They might be a bit 'iffy' about the introduction of a foreign supply into their network! But this isn't a permanent solution we're talking about of course.
Oh, no.
As far as the POTS element of telephony is concerned, I've had no experience of repairs or faults other than by BT Openreach.