Long-term rentals everywhere around the world typically require the tenant to prove the ability to pay the current and every future month rent. This is done by showing salary slips (usually the last 3 in France), payment receipts from previous rental agreements, a reference letter, and/or binding someone else to be your guarantor to cover any missed payments (this person too will have to show proofs). The paperwork required hence has nothing to do with citizenship or nationalism. Rather it's simply pure business.
This explains why short-term rental services such as airbnb have become widely popular recently: the tenant pays the money upfront upon booking, and the landlord receives it 24h after the tenant moves in. Therefore I would suggest you looked into airbnb listings for Cahors to get a rough idea of the prices. Keep in mind that they will inevitably be more expensive than those of long-term rentals.
There's another interesting category of short-term rentals, namely holiday rentals. These are usually spare houses in locations which might attract tourists (seaside, mountains, etc). More recently, people have started renting their own house in the months they won't be using it. In French this service would be known as
Location Vacances. You can use Roomlala (previously known as Chambre à Louer) to find such type of accomodation in Cahors (see here for some listings).
Since you don't mention whether you want an entire flat or a flat share (
Colocation in French) here is another suggestion. For the latter, have a look at leboncoin listings for Cahors (leboncoin is a French version of Gumtree). Note that you can also find long-term rentals there. It might be worth contacting landlords anyway to ask for short-term (listings from particuliers (= privates) as opposed to professionels (= agencies), is what you want).
Last but not least, another way to find short-term rental flats is to lookout for students. Usually these pay for 12-months rent and end up using less since they eventually leave for summer holidays. I imagine they would be happy to sublet the months they wouldn't be using so as to not lose any money. You might be in luck since there's a university in Cahors. Unfortunately, the only way to find these offers is either via leboncoin listings, or by checking the various bulletin boards spread across universities, supermarkers, bakeries, etc.
Rent a Room
Should you be unable to find a flat, you can also think about renting a room in a house. In France this would be a
Chambre chez l'Habitant. Roomlala can also be used to look for such type of accomodation in Cahors (see here for some listings).