For the kind of flights/costs you are requesting, I'm afraid the FA50 will be your only chance.
You may try to call La Mole airport to check what kind of aircrafts are approved.
I do fly to La Mole with a FA50B, and if you strictly comply with the performances you're limited to 2 pax in commercial ops....(that's the only way we use it)
You should check about private ops with a FA50B and FA50EX.
I guess the 7X is not an option for you......(too bad cause, if I'm right, it could fly non-stop to New-York from La Mole....)
The Falcon50 is also London-City approved ....
Jetopa said :"I hear that the Gulfstreams have excellent runway performance", I do not know where he heard that but I've never seen any Gulfstream at La Mole airport neither at London City (except for the G150, but is this really a Gulfstream ....?)
Good luck ...
Christophe.