  We apply Lescop's construction of $\mathbb{Z}$-equivariant perturbative
invariant of knots and 3-manifolds to the explicit equivariant propagator of
"AL-paths" given in arXiv:1403.8030. We obtain an invariant $\hat{Z}_n$ of
certain equivalence classes of fiberwise Morse functions on a 3-manifold
fibered over $S^1$, which can be considered as a higher loop analogue of the
Lefschetz zeta function and whose construction will be applied to that of
finite type invariants of knots in such a 3-manifold. We also give a
combinatorial formula for Lescop's equivariant invariant $\mathscr{Q}$ for
3-manifolds with $H_1=\mathbb{Z}$ fibered over $S^1$. Moreover, surgery
formulas of $\hat{Z}_n$ and $\mathscr{Q}$ for alternating sums of surgeries are
given. This gives another proof of Lescop's surgery formula of $\mathscr{Q}$
for special kind of 3-manifolds and surgeries, which is simple in the sense
that the formula is obtained easily by counting certain graphs in a 3-manifold.
