As far as I know, a foreigner can register a car in their name in France if they own a property here, so buying a mobile home is one, if rather expensive solution, although there are probably quite cheap units for sale in less desirable areas.
EDIT: A further problem is the high cost of taxes on buying and selling property in France, about 10%, more on cheap properties. Check this with a Notaire, or on line if you understand French.
Another possibility is to buy a (preferably LHD) camper in the UK, where registration problems are not so difficult.
If you were to do this, you should advise the authorities of its export, to avoid liability to UK road tax while it is out of the country, by completing and mailing the tear-off slip on the licence document (V5C). Customs on return to UK can inform you on how to reinstate the UK registration.
There are various companies offering full insurance cover on the Continent for longer than the usual 30 days, but some of these require the driver to have a UK driving licence. Googling "european insurance for uk car" will provide more information on this.
On some great and glorious day the plain folks of the land will reach their heart's desire at last, and the White House will be adorned by a downright moron.
H. L. Mencken 1880 - 1956
Some may not like his views, but what a prediction!