  Measurements of nuclear $\beta$-decay are commonly used to extract elements
of the quark-mixing Cabibbo-Kobayashi-Maskawa (CKM) matrix. The precision of
these measurements is currently limited by theoretical uncertainties in
electroweak radiative corrections (EWRC) arising from so-called box diagrams,
which involve the exchange of two gauge bosons. Two photon exchange (TPE) is
the most experimentally accessible of such processes, making it a natural
choice for providing constraints on the theoretical frameworks used for
calculating EWRC. The cross section asymmetry between elastic $e^+/e^-$
scattering is directly sensitive to this, as the TPE contribution to the cross
section has opposite sign for electrons and positrons. While charge asymmetry
measurements have been performed on proton targets, no such data exists for
nuclear targets. Proposed here is a measurement of the $e^+/e^-$ charge
asymmetry on various nuclei relevant to $\beta$-decay measurements used for CKM
matrix extraction. Determining the size of TPE through these processes can
provide important constraints on EWRC.
