Assuming that SCYSA does not make a special adjustment to the current seasonal year cup manual....
If Bridge 93 plays in RIII as a U14, then no they could not play as a 93 team in the cup, without playing U13 in the SCSCL also. Interesting point would be the seeding for the U14 should that happen.
The team that plays in the spot has to have the roster integrity described in the manual. Depending on the number of players rostered on the team when the spot was earned a certain number of players is required. In general, the spot is earned by the team, not the club.
SCYSA does have the ability to select it's spots, but also has the duty to put competitive teams forward.
Historically speaking 3rd place teams from SC in the RIII haven't proven as competitive as 3rd place teams from NC or GA. I'm not sure the SCYSA would put a SC team in if they didn't feel they would be competitive.