Here is something in answe to myself. I'm not 100% sure it's correct, and I don't know yet if it can be used to find concrete examples. Let $K_{n}$ be the intersection of all normal subgroups of $F$ of index $n$. Then $\hat{F}$ is the inverse limit of $F/K_{n}$, and the automorphism group of $\hat{F}$ is the inverse limit of automorphism groups of these quotients. Hence thw closure of $H\subset F$ is characteristic in $\hat{F}$ iff the image of $H$ is characteristic in $F/K_{n}$ for every $n$,

@YCor The linked question considered general $F$, I was hoping that restricting to free $F$ might help to gather new answers.