Im not sure if that is the same. Jane F is back in her home town, where she is a well known and loved figure and it si a tiny place where all people of a certian rank socialise.. which annoys Emma.. in that it is very hard to preserve one's "exclusivity" in such a small place.
But Jane B has gone to London.. she is in a different part of London to the Bingleys and is not moving in their circles. Bing does not know the Gardiners, so I think that it would be a bit different for him to pay tehm a call. It would look like he was doing it largely to get in touch with Jane.. so unless he was darned sure he wanted to renew his relationship with J at a serious level, I dont think he'd be expected to make a "duty call".
I think that it is more likely that if Jane and Caro met up and Bing knew that she was in town, he would say to Caroline to invite Jane and her aunt and uncle to dinner or a party, and take it from there. So that is why Caro and D had to conceal Jane's being in town from him.