Compact Schnorr signature?
This would work, but note that your weakening verification (slightly). Instead of only $(C_1,k_2)$ being a valid signature, now also $(-C_1,k_2)$ is valid. If you want to do signatures by only using $x$-coordinates you can use the qDSA signature scheme, which indeed saves a few bits in the signature size and public key.

There is actually a more serious compression of Schnorr signatures, which I think is quite common. It is already described in Schnorr's original paper. Instead of having a signature $(C_1,k_2)$ we use $(e,k_2)$, where it actually suffices to have relatively small $e$ (say 128 bits for 128-bit security level). Now we can verify by computing $\bar C_1 = [k_2]G - [e]C$ and checking that
$H(\bar C_1\,||\,M) = e$. 

This leads to a 48-byte signature as opposed to a 64-byte one, so saves significantly more than a bit. A downside is that you cannot batch-verify signature anymore, though I don't know if that's actually used in practice.