  We perform perturbative computations in a lattice gauge theory with a
conformal measure that is quadratic in a non-compact abelian gauge field and is
nonlocal, as inspired by the induced gauge action in massless QED$_3$. In a
previous work, we showed that coupling fermion sources to the gauge model led
to nontrivial conformal data in the correlation functions of fermion bilinears
that are functions of charge $q$ of the fermion. In this paper, we compute such
gauge invariant fermionic observables to order $q^2$ in lattice perturbation
theory with the same conformal measure. We reproduce the expectations for
scalar anomalous dimension from previous estimates in dimensional
regularization. We address the issue of the lattice regulator dependence of the
amplitudes of correlation functions.
