The earliest a flight can land and LCY on a Sunday is 12.30 PM. The latest it can take off on a Saturday is 1pm - And a departure time would have to also fit in with the operating times of the SNN pre clearance otherwise what would be the point.
As to Billy Bishop Airport - there was a proposal from Porter to use CS100 Jets (the same as earlier mentioned that can and does operate from LCY) but that needed a 5-600m lengthening of the runway and Porter went about proposing that in such a cack handed way that it basically sunk it's own proposal.
And as for suggestions to have it land ot LGA or other NY State airports - why would BA want to open yet another base for a single flight a day with all the costs associated with that and likely making it an even less premium service.