  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 4630-50\AA. 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 $M_{\odot}$).
