Phase-resolved VLT spectroscopy of the bursting low-mass X-ray binaries 4U 1636-536/V801 Ara and 4U 1735-444/V926 Sco is presented. Doppler images of the NIII λ4640 Bowen transition reveal compact spots which we attribute to fluorescent emission from the donor star and enable us to define a new set of spectroscopic ephemerides. We measure Kem = 277 +/- 22 and 226 +/- 22kms-1 from the NIII spots in V801 Ara and V926 Sco, respectively, which represent strict lower limits to the radial velocity semi-amplitude of the donor stars. Our new ephemerides provide confirmation that light-curve maxima in V801 Ara and likely V926 Sco occur at superior conjunction of the donor star and hence photometric modulation is caused by the visibility of the X-ray heated donor. The velocities of HeII λ4686 and the broad Bowen blend are strongly modulated with the orbital period, with phasing supporting emission dominated by the disc bulge. In addition, a reanalysis of burst oscillations in V801 Ara, using our spectroscopic T0, leads to K1 = 90-113kms-1. We also estimate the K-corrections for all possible disc flaring angles and present the first dynamical constraints on the masses of these X-ray bursters. These are K2 = 360 +/- 74kms-1, f(M) = 0.76 +/- 0.47Msolar and q = 0.21-0.34 for V801 Ara and K2 = 298 +/- 83kms-1,f(M) = 0.53 +/- 0.44Msolar and q = 0.05-0.41 for V926 Sco. Disc flaring angles α >= 12° and q ~= 0.26-0.34 are favoured for V801 Ara whereas the lack of K1 constraint for V926 Sco prevents tight constraints on this system. Although both binaries seem to have intermediate inclinations, the larger equivalent width of the narrow NIII line in V801 Ara at phase 0.5 relative to phase 0 suggests that it has the higher inclination of the two.