Use $e$ in GGH as shared secret?
That wouldn't appear to work.  In a known plaintext attack, that is, where the attacker knows both $m$ and $c$, he can compute:

$$e = c - m * B$$

recovering the secret key.

Alternatively, in a ciphertext-only scenario, the attacker could take two ciphertexts $c, c'$ and compute

$$(c - c') * B^{-1} = m - m'$$

resulting in the difference between the two plaintexts.