If FGW truly wanted to charter aircraft to operate air service for 6 weeks until the trains are (supposedly) going to be back up an running then of course it doesn't have to be flyBe they use.
This time of year it isn't going to be too difficult to find spare aircraft and crew (yes even ones with LCY clearances) to cover that.
Cityflyer - probably not, but I can immediately think of another good 2-3 carriers with LCY-trained crews that would probably be able to provide an aircraft for a 2 or 3 daily service for this period..