The exocometary origin of CO gas has been confirmed in several extrasolar Kuiper belts, with CO ice abundances consistent with Solar System comets. We here present a molecular survey of the $\beta$ Pictoris belt with the Submillimeter Array (SMA), reporting upper limits for CN, HCN, HCO$+$, N$2$H$+$ and H$2$CO, as well as for H$2$S, CH$3$OH, SiO and DCN from archival ALMA data. Non-detections can be attributed to rapid molecular photodissociation due to the A-star's strong UV flux. CN is the longest-lasting and most easily detectable molecule after CO in this environment. We update our NLTE excitation model to include UV fluorescence, finding it plays a key role in CO and CN excitation, and use it to turn the SMA CN/CO flux ratio constraint into an upper limit of % on the HCN/(CO+CO$2$) ratio of outgassing rates. This value is consistent with, but at the low end of, the broad range observed in Solar System comets. If sublimation dominates outgassing, then this low value may be caused by decreased outgassing for the less volatile molecule HCN compared to CO. If instead UV photodesorption or collisional vaporization of unbound grains dominates outgassing, then this low ratio of rates would imply a low ice abundance ratio, which would in turn indicate a variation in cometary cyanide abundances across planetary systems. To conclude, we make predictions for future molecular surveys and show that CN and HCN should be readily detectable with ALMA around $\beta$ Pictoris for Solar-System-like exocometary compositions.