  The goal of this paper is to investigate the Theta invariant --- an invariant
of framed 3-manifolds associated with the lowest order contribution to the
Chern-Simons partition function --- in the context of the quantum BV-BFV
formalism. Namely, we compute the state on the solid torus to low degree in
$\hbar$, and apply the gluing procedure to compute the Theta invariant of lens
spaces. We use a distributional propagator which does not extend to a
compactified configuration space, so to compute loop diagrams we have to define
a regularization of the product of the distributional propagators, which is
done in an \emph{ad hoc} fashion. Also, a polarization has to be chosen for the
quantization process. Our results agree with results in the literature for one
type of polarization, but for another type of polarization there are extra
terms.
