I am also of the view that if the OP wants to resolve this then contacting the local Mairie, perhaps with a view to them contacting the local police/Gendarmes (as appropriate, depends on villa location) might be the way to go. I have heard of police/Gendarmes doing spot checks on holiday homes.
FWIW being a holiday home (i.e. second home owner) is not a reason for entry ATM.
To enter France currently you need to produce a completed " attestation de déplacement international."
It has various tick boxes where you indicate which category you fall in , e.g. French national, Individuals having their primary residence in France...on arrival you can expect Border Police to ask for the form and supporting documentation (e.g. professional ID).
As has been said "Essential business" in itself, on it's own, doesn't qualify though I can see some categories where it might be possible to blag entry if you are carrying a convincing letter:
This will make for a long post, but FWIW here's the full list as it applies to Brits the last time I checked:
" For European Union or assimilated countries nationals:
Individuals having their primary residence in France, accompanied by their spouse and children;
Individuals transiting through France to reach their residence, accompanied by their spouse and children;
Healthcare workers supporting the fight against Covid-19;
Goods carriers, including seamen;
Flight and cargo crews, or travelling as a passenger to their departure base;
Diplomatic mission staff, or international organisations staff working in headquarters or offices located in France,
who are holders of a special residence permit or a type D promae visa;
Cross-border workers at internal land borders.
French nationals, accompanied by their spouse and children"