A comprehensive study of the mass sensitivity of the vibration-rotation-inversion transitions of $^{14}$NH$_3$, $^{15}$NH$_3$, $^{14}$ND$_3$, and $^{15}$ND$_3$ is carried out variationally using the TROVE approach. Variational calculations are robust and accurate, offering a new way to compute sensitivity coefficients. Particular attention is paid to the $Δ k=± 3$ transitions between the accidentally coinciding rotation-inversion energy levels of the $ν_2=0^+,0^-,1^+$ and $1^-$ states, and the inversion transitions in the $ν_4=1$ state affected by the "giant" $l$-type doubling effect. These transitions exhibit highly anomalous sensitivities, thus appearing as promising probes of a possible cosmological variation of the proton-to-electron mass ratio $μ$. Moreover, a simultaneous comparison of the calculated sensitivities reveals a sizeable isotopic dependence which could aid an exclusive ammonia detection. ; Comment: 18 pages, 2 figures, 10 tables