There seems to be a situation here in which the right hand is not supposed to know what the left hand is doing. To keep Meryton gossip in Meryton so far as is possible. The situation is not thorough, as people here and there (including a regiment in Brighton) have at least some knowledge of the facts. Certainly the Old Ladies of Meryton have enough to gossip about.

My take is that Lydia's marriage needed to be recognized at Longbourn primarily for her sisters' sake (which was potentially also in Darcy's interest even this early in the story). I guess one today might call what they were doing as creating "deniability."

But not to acknowledge the Longbourn connection in the London papers was presumably designed to minimize the chance that the Meryton gossip would easily reach London (to both Darcy's and the Gardiners' benefit) or Newcastle (for the Wickhams' sake, for what it will be worth) or Pemberley. (Sadly keeping it from Rosings was out of the question.)