The nature of GW190425, a presumed binary neutron star (BNS) merger detected by the LIGO/Virgo Scientific Collaboration (LVC) with a total mass of $3.4^{+0.3}_{-0.1}$ M$_{\odot}$, remains a mystery. With such a large total mass, GW190425 stands at five standard deviations away from the total mass distribution of Galactic BNSs of $2.66\pm 0.12$ M$_{\odot}$. LVC suggested that this system could be a BNS formed from a fast-merging channel rendering its non-detection at radio wavelengths due to selection effects. BNSs with orbital periods less than a few hours - progenitors of LIGO/Virgo mergers - are prime target candidates for the future Laser Interferometer Space Antenna (LISA). If GW190425-like binaries exist in the Milky Way, LISA will detect them within the volume of our Galaxy and will measure their chirp masses to better than 10% for those binaries with gravitational wave frequencies larger than 2 mHz. This work explores how we can probe a population of Galactic GW190425-like BNSs with LISA and investigate their origin. We assume that the Milky Way's BNS population consists of two distinct sub-populations: a fraction $w_1$ that follows the observed Galactic BNS chirp mass distribution and $w_2$ that resembles chirp mass of GW190425. We show that LISA's accuracy on recovering the fraction of GW190425-like binaries depends on the BNS merger rate. For the merger rates reported in the literature, $21 - 212\,$Myr$^{-1}$, the error on the recovered fractions varies between $\sim 30 - 5$%.