The gravitational collapse of rapidly rotating massive stars can lead to the onset of the low $T/\|W\|$ instability within the central proto-neutron star (PNS), which leaves strong signatures in both the gravitational wave (GW) and neutrino emission. Strong large-scale magnetic fields are usually invoked to explain outstanding stellar explosions of rapidly rotating progenitors, but their impact on the growth of such instability has not yet been cleared. We analyze a series of three-dimensional magnetohydrodynamic models to characterize the effects of different magnetic configurations on the development of the low $T/\|W\|$ and the related multi-messenger features. In the absence of magnetic fields, we observe the growth on dynamical time scales of the low $T/\|W\|$, associated with a strong burst of GW and a correlated modulation of the neutrino emission. However, models with a strong magnetic field show a quenching of the low $T/\|W\|$, due to a flattening of the rotation profile in the first $\sim100$ ms after shock formation caused by the magnetic transport of angular momentum. The associated GW emission is weakened by an order of magnitude, exhibits a broader spectral shape, and has no dominant feature associated with the PNS large-scale oscillation modes. Neutrino luminosities are damped along the equatorial plane due to a more oblate PNS, and the only clear modulation in the signal is due to SASI activity. Finally, magnetized models produce lower luminosities for $\nu_e$ than for $\bar{\nu}_e$, which is connected to a higher concentration of neutron-rich material in the PNS surroundings.