I just figured that FD would assume that CF wouldn't give any info (or as little as politely possible) about Georgiana -Ramsgate particularly! - unless FD had spoken to him beforehand. And the fact that he says in his letter "that there may be the possibility of consulting him, I shall endeavour to find some opportunity of putting this letter in your hands in the course of the morning makes me think FD thinks it is possible she will want to consult CF. So I doubt he'd leave it to chance.

On the other hand, I don't think he'd want to share the details of his (brutal!) rejection with anyone so your explanation seems likely.