  In the post-Newtonian parametrization of semi-conservative gravity theories,
local Lorentz invariance (LLI) violation is characterized by two parameters,
alpha_1 and alpha_2. In binary pulsars the isotropic violation of LLI in the
gravitational sector leads to characteristic preferred frame effects (PFEs) in
the orbital dynamics, if the barycenter of the binary is moving relative to the
preferred frame with a velocity w. For small-eccentricity binaries, the effects
induced by alpha_1 and alpha_2 decouple, and can therefore be tested
independently. We use recent timing results of two compact pulsar-white dwarf
binaries with known 3D velocity, PSRs J1012+5307 and J1738+0333, to constrain
PFEs for strongly self-gravitating bodies. We derive a limit |alpha_2| < 1.8e-4
(95% CL), which is the most constraining limit for strongly self-gravitating
systems up to now. Concerning alpha_1, we propose a new, robust method to
constrain this parameter. Our most conservative result, alpha_1 =
-0.4^{+3.7}_{-3.1} e-5 (95% CL) from PSR J1738+0333, constitutes a significant
improvement compared to current most stringent limits obtained both in Solar
system and binary pulsar tests. We also derive corresponding limits for alpha_1
and alpha_2 for a preferred frame that is at rest with respect to our Galaxy,
and preferred frames that locally co-move with the rotation of our Galaxy.
(Abridged)
