The LIGO-Virgo-KAGRA Collaboration recently detected gravitational waves (GWs) from the merger of black hole-neutron star (BHNS) binary systems GW200105 and GW200115. No coincident electromagnetic (EM) counterparts were detected. While the mass ratio and BH spin in both systems were not sufficient to tidally disrupt the NS outside the BH event horizon, other, magnetospheric mechanisms for EM emission exist in this regime and depend sensitively on the NS magnetic field strength. Combining GW measurements with EM flux upper limits, we place upper limits on the NS surface magnetic field strength above which magnetospheric emission models would have generated an observable EM counterpart. We consider fireball models powered by the black hole battery mechanism, where energy is output in gamma rays over less than or similar to 1 s. Consistency with no detection by Fermi-GBM or INTEGRAL SPI-ACS constrains the NS surface magnetic field to less than or similar to 10(15) G. Hence, joint GW detection and EM upper limits rule out the theoretical possibility that the NSs in GW200105 and GW200115, and the putative NS in GW190814, retain dipolar magnetic fields greater than or similar to 10(15) G until merger. They also rule out formation scenarios where strongly magnetized magnetars quickly merge with BHs. We alternatively rule out operation of the BH-battery-powered fireball mechanism in these systems. This is the first multimessenger constraint on NS magnetic fields in BHNS systems and a novel approach to probe fields at this point in NS evolution. This demonstrates the constraining power that multimessenger analyses of BHNS mergers have on BHNS formation scenarios, NS magnetic field evolution, and the physics of BHNS magnetospheric interactions.