Yes, supposedly.
SF would just play musical chairs. If Richmond were to go to AAA without any expansion, some city would drop to AA.
It would have to be in or near one of the AA league footprints (as in, not west of Texas or anywhere in the upper midwest). There's not really an obvious candidate between geographical fit, ownership situation (for example, Syracuse and Rochester would be semi-logical candidates but both have ownership that would make it unworkable), or affiliation.
On the merits, there is no reason there should be a AAA team in the Atlanta suburbs (equally so for a AA team) and the team draws poorly despite a great facility and is owned by a PE firm that is very open to both moves and level swaps (having done both in the past year), but there's no way the Giants or Braves would want SF's AA team to be in the Atlanta equivalent of Herndon.
Barves AA is in Pearl, Mississippi. I don't know if that would be a location the Giants would want, but perhaps you could see something like this:
Giants move AA from Richmong to Pearl.
Braves move AA from Pearl to Gwinett.
Gwinett's AAA affiliation moves to Richmond.
Nats drop Rochester and affiliate with Richmond.
Braves affiliate with Rochester.
I don't see the Braves being that charitable to the Nats. I would think that if teir AAA is not in Gwinett, they'd love to be back in a good ballpark in Richmond.
I actually think the As to Vegas will be the next reshuffle. Really would only need a new AA city in the East and an IL team that would shift to the PCL, along with an MLB team currently with a PCL affiliate that prefers Rochester.