We present phase-resolved spectroscopy of the millisecond X-ray pulsar XTE J1814-338 obtained during its 2003 outburst. The spectra are dominated by high-excitation emission lines of HeII $lambda$4686, H$beta$, and the Bowen blend CIII/NIII 4630-50AA. We exploit the proven Bowen fluorescence technique to establish a complete set of dynamical system parameter constraints using bootstrap Doppler tomography, a first for an accreting millisecond X-ray pulsar binary. The reconstructed Doppler map of the NIII $lambda$4640 Bowen transition exhibits a statistically significant (>4$sigma$) spot feature at the expected position of the companion star. If this feature is driven by irradiation of the surface of the Roche lobe filling companion, we derive a strict lower limit to the true radial velocity semi-amplitude $K_2$. Combining our donor constraint with the well constrained orbit of the neutron star leads to a determination of the binary mass ratio: q = $0.123^{+0.012}_{-0.010}$. The component masses are not tightly constrained given our lack of knowledge of the binary inclination. We cannot rule out a canonical neutron star mass of 1.4 $M_{odot}$ (1.1 $M_{odot}$ < $M_1$ < 3.1 $M_{odot}$; 95%). The 68/95% confidence limits of $M_2$ are consistent with the companion being a significantly bloated, M-type main sequence star. Our findings, combined with results from studies of the quiescent optical counterpart of XTE J1814-338, suggest the presence of a rotation-powered millisecond pulsar in XTE J1814-338 during an X-ray quiescent state. The companion mass is typical of the so-called redback pulsar binary systems ($M_2$ ~ 0.2 $M_{odot}$).
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