What is an approximate length for a digital signature (SHA) in this case?
BLS signatures are $\:2\hspace{-0.04 in}\cdot\hspace{-0.03 in}k\:$ bits long, where $k$ is the security parameter, and the probability of a forgery is $\hspace{.01 in}\epsilon$.

Pseudorandom MACs (such as HMAC) can be truncated to $L\hspace{.01 in}$ bits, and the probability

of forgery (by someone who is outside of the system) will be $\: \frac{\text{# of tries}}{2^{\hspace{.01 in}L}}+\epsilon \;\;$.

(As alluded to in my comment, the verifier(s) can trivially forge valid authenticators for the MAC approach.)