The PADME experiment at Laboratori Nazionali di Frascati (LNF), Italy, will search for invisible decays of the hypothetical dark photon via the process $e^+ e^- \rightarrow \gamma A’$, where the $A’$ escapes detection. The dark photon mass range sensitivity will be 1 to 24 MeV at first. We report here on performance measurements and simulation studies of a prototype of the Small-Angle Calorimeter, a component of PADME’s detector dedicated to rejecting 2-$\gamma$ and 3-$\gamma$ backgrounds. The crucial requirement is a timing resolution of less than 200 ps, which is satisfied by the choice of $PbF_2$ crystals and the newly released Hamamatsu R13478UV photomultiplier tubes (PMTs). We find a timing resolution of 81 ps with double-peak separation resolution of 1.8 ns and a single-crystal energy resolution of 10% at 550 MeV with light yield of 2.05 photo-electrons per MeV, using 100 to 400 MeV electrons at the Beam Test Facility of LNF. We also propose the investigation of a two-PMT solution coupled to a single $PbF_2$ crystal for higher-energy applications, which has potentially attractive features.