Zachary. You mention neutrinos hitting electrons. In the electromagnetic interaction, a high energy photon can hit an electron and speed it up...Compton scattering....and the photon is redshifted. But neutrinos do not participate in the electromagnetic interaction or the strong interaction, only the weak interaction. Here, they can act via the two charged currents W⁺, or W^-, or the neutral current Z⁰. The emission of any of the three will redshift the neutrino, so it "looks" like the same thing externally, but the carrier in the Compton scatter is a photon, so they're distinct.
Yes the white dwarf can pick up some neutrino energy, and a subtle effect might show with a periodicty related to the revolution period. It seems from the SNO day/night periodicity that the Earth is sufficiently massive to show a ~14% change in reaction rates, which through scattering would represent ~ a 10 % energy drop for scattered neutrinos. Say the central star emitted 10 Mev neutrinos. If each scattered once, that would drop their average energy to ~ 9 Mev (James Losecco,Head, Dept. of Physics Univ. Notre Dame). The neutrino cross-section varies as the ~ square of the energy (Ultimate Neutrino Page)...so 9² compared to 10²...is 81 to 100. Roughly an order of magnitude answer.
Your dwarf only intercepts a small fraction of the central companion's neutrinos though, and you need to approximate it's subtended angle in the sky using an assumption of isotropic emission (not quite true, but will suffice for OOM calculations). ....and remember you have a sphere, not a disk.
It will not have much effect on a young white dwarf, as the energy emission scales as (Kelvin)^4thbut will contribute to a cooling anomaly in an old cold one....the dang thing will seem to be fusing slowly. pete

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A third rate theory forbids
A second rate theory explains after the fact
A first rate theory predicts...A. Lomonosov