Persistent, low-luminosity low-mass X-ray binaries offer a unique opportunity to study accretion in this poorly understood regime as well as to unveil new members of the ultracompact X-ray binary (UCXB) family, which are characterised by orbital periods (Porb) shorter than ∼80 min. We report on a NuSTAR archival observation that, for the first time above 10 keV, spatially resolves the Galactic Centre pair SLX 1744−299 and SLX 1744−300. We find SLX 1744−300 to be slightly brighter, with a flux ratio of ∼1.15, increasing to ∼1.3 when extrapolated to 0.5─10 keV. Both the timing (root-mean-square variability) and spectral properties (well described in both cases by a thermal Comptonisation model) indicate that the systems were in the hard state. The two sources, however, display markedly different behaviour throughout the observation. SLX 1744−299 shows a gradual flux decline consistent with a decrease in the mass-accretion rate, whereas SLX 1744−300 remains steady but exhibits two short-recurrence Type-I X-ray bursts indicative of mixed H and He burning. Combining our results with previously reported upper limits on the distance, we derived low persistent X-ray luminosities of LX ≲ 1.1 × 1036 erg s−1 and LX ≲ 2.6 × 1036 erg s−1 (3─78 keV) for SLX 1744−299 and SLX 1744−300, respectively. The corresponding mass-accretion rates, when compared with the critical values from the disc instability model, favour Porb ≲ 90 min and Porb ≲ 105─155 min. Although both limits are formally compatible with the UCXB regime, the case of SLX 1744−299 appears significantly more compelling, also considering the previously reported intermediate-duration burst.