  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.
