You're right: If the FCC stops at London Bridge on the way to Gatwick, then you will definitely be able to get on it and go. And the way that London Bridge station is set up, if you come from Charing Cross then you will almost certainly have an easy same-platform or cross-platform change like on the way into town.

The difficulty, if there is one, is that you may end up travelling just in the "hole" in the FCC service. FCC departures on weekday mornings from London Bridge to Gatwick are at:-
0644 (and earlier)
0700
0716
0740
0820
0905
0925
0941
0955 (and later)

So you see that the 15-minute pattern that is in operation through most of the day is disrupted during the morning peak. This is, I believe, because London Bridge does not have enough capacity to deal with all the FCC trains, and gives priority to those which have to use those platforms of London Bridge because they are terminating at Cannon Street or Charing Cross. The FCC trains can be diverted through a more "scenic" route so their first central London station is Blackfriars.

Nevertheless, there are still plenty of trains from London Bridge to Gatwick during the morning peak. It may just be a little more hassle, as I say, because you could have to change platforms at London Bridge. But I doubt that this would be any more inconvenient than changing from Tube to train at Victoria. If you're lucky, you could end up with a same- or cross-platform change anyway, from getting on one of the FCC trains that still stops at London Bridge.