Avnu
The BLL and HAM routes are 100% SunAir's own choice and own risk.
They would technically have to ask BA (mainline, with whom they hold the franchise) for approval to operate the routes under said franchise - in turn no doubt BA would check with CityFlyer if this was likely to cause any issues...
Since CityFlyer never showed any interest in these routes themselves (indeed BA mainline going on LHR-BLL will have been a kick in the teeth for SunAir) and since they seem keen to lock up as many peak LCY slots as possible under BA-branded operations, I'd have said they'd have positively welcomed SunAir coming to do these routes..