The bad news is that your UK air rifle is well over the limit, power-wise for France. This means it will be classified as a 'declarable' weapon which will require a British FAC and an International Firearms Permit in order to move it around inside the EU.
Once in France (You
WON'T be flying will you - with a
RIFLE! ) you will have to register it with the local Gendarmerie...in French and in person.....with your passport and all your UK paprwork, including bill of sale to prove ownership.
Getting it back again past YOUR customs should be a laugh!
Once here, you still cannot legally hunt for 2 reasons: (1) Hunting with air rifles is illegal in France and (2) You need a hunting permit (Permis de Chasse) for which you have to pass a test.
In short, it's not worth the hassle.
The good news is that this is France. Depending on how rural your location is going to be, you may find that a local Farmer will be quite happy to lend you a battered rimmie if you're going to be popping off his pigeons (They don't shoot corvids - they are of the opinion they do a useful job) and rats (as long as you buy the ammo - which you can do with no paperwork).
If it's VERY rural, you may even get the local cop asking for a go.......!
Hope that helps.