  Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$
with values in the space of differential operators acting on sections of some
bundles, by means of the Finsler function $F.$ As an operator, it has several
expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection,
although its cohomology class does not depend on them. This cocycle is closely
related to the conformal Schwarzian derivatives introduced in our previous
work. The second main result of this paper is to discuss some properties of the
conformally invariant quantization map by means of a Sazaki (type) metric on
the slit bundle $TM\backslash 0$ induced by $F.$
