OK, final thoughts on this for the original query.
You need to check as to whether the Commune has changed since the birth date that you have. My commune was much much bigger back then and it has now been split into several smaller Communes that have themselves been adapted and amended so the records may be in a different Commune. As the record isn't online, I would write to the Mairie of the Commune concern and throw yourself on their mercy and ask them to do a search for you - also ask them if any of the records are missing at all.
Additionally, if you do decide to apprach the parish, be aware that in rural and semi-rural areas, parishes are often covered by one priest who may cover up to 8 parishes, which themselves may have changed over the years and what was in one parish then may not be the same parish now. And do not assume that the priest or anybody else will help, that does not always happen in France where genealogy can come quite low on the totem pole in rural areas.
The above only comes from years of practical, hands-on research in French country areas so hope it helps.
Regards
TonyF