  The presence of a companion wind in neutron star binary systems can form a
contact discontinuity well within the pulsar's light cylinder, effectively
creating a waveguide that confines the pulsar's electromagnetic fields and
significantly alters its spindown. We parametrize this confinement as the ratio
between the equatorial position of the contact discontinuity (or standoff
distance) $R_\mathrm{m}$ and the pulsar's light cylinder $R_\mathrm{LC}$. We
quantify the pulsar spindown for relativistic wind envelopes with
$R_\mathrm{m}/R_\mathrm{LC} = 1/3...1$ and varying inclination angles $\chi$
between magnetic and rotation axes using particle-in-cell simulations. Our
strongly confined models ($R_\mathrm{m}/R_\mathrm{LC} = 1/3$) identify two
distinct limits. For $\chi=0^\circ$, the spindown induced by the compressed
pulsar magnetosphere is enhanced by approximately three times compared to an
isolated pulsar due to an increased number of open magnetic field lines.
Conversely, for $\chi=90^\circ$, the compressed system spins down at less than
$40\%$ of the rate of an isolated reference pulsar due to the mismatch between
the pulsar wind stripe wavelength and the waveguide size. We directly apply our
analysis to the 2.77-second period oblique rotator ($\chi=60^\circ$) in the
double pulsar system PSR J0737-3039. With the numerically derived spindown
estimate, we constrain its surface magnetic field to $B_* \approx (7.3 \pm 0.2)
\times 10^{11}$ G. We discuss the time modulation of its period derivative, the
effects of compression on its braking index, and implications for the radio
eclipse in PSR J0737-3039.
