2-connected isn't entirely standard - I've also seen `2-linked' used for the concept of 2-connected described here. You're right that 2-connected doesn't imply connected here. As to an approach, I tried (following the lead of Sapozhenko) to find a small set of subsets which appropriately approximate any given 2-connected set of size a with neighborhood of size n, and then for each approximating set I tried to bound the number of a that could approximated by this particular approximating set. This leads to the $2^{n - cn/\log d}$ result.

Sorry - I edited the post. I'm interested in $2^{d/\log^2 d} \leq a \leq c^d$ (for some fixed $1 < c < 2$), and I'm looking for asymptotics in $d$.

Sorry - I didn't read the 'exists' part of your question carefully, so this isn't quite what you're looking for. It is a bound for a particular $n$, though.