I know that not all truthtables have a corresponding 2-cnf representation, but is there a way to find out if a given truthtable has a 2-cnf representation, and if so to find what that is?
This answer assumes that a 2CNF representation of a function is a 2CNF (on the same set of variables) that agrees with the function on all inputs.
Let's say that a clause $C$ is consistent with a function $f$ if $\lnot C \Rightarrow \lnot f$. Let $C_1,\ldots,C_m$ be the collection of 2-clauses consistent with your function $f$. Then $f$ has a 2CNF representation if and only if $f = C_1 \lor \cdots \lor C_m$. This gives a quasilinear time algorithm for deciding whether $f$, given as a truth table, has a 2CNF representation.