The radionuclide $^{22}$Na is a target of $\gamma$-ray astronomy searches, predicted to be produced during thermonuclear runaways driving classical novae. The $^{22}$Na(p,$\gamma$)$^{23}$Mg reaction is the main destruction channel of $^{22}$Na during a nova, hence, its rate is needed to accurately predict the $^{22}$Na yield. However, experimental determinations of the resonance strengths have led to inconsistent results. In this work, we report a measurement of the branching ratios of the $^{23}$Al $\beta$-delayed protons, as a probe of the key 204--keV (center-of-mass) $^{22}$Na(p,$\gamma$)$^{23}$Mg resonance strength. We report a factor of 5 lower branching ratio compared to the most recent literature value. The variation in $^{22}$Na yield due to nuclear data inconsistencies was assessed using a series of hydrodynamic nova outburst simulations and has increased to a factor of 3.8, corresponding to a factor of $\sim$2 uncertainty in the maximum detectability distance. This is the first reported scientific measurement using the Gaseous Detector with Germanium Tagging (GADGET) system.
