French cheques?

I am guessing they just literally mean a check drawn on a French banking account. I don't think you have any way of obtaining one.

Ask them for their bank account details and transfer the amount directly

Americans don’t have access to that bank transfer system without paying an overseas transfer wire to their banks, unfortunately.

My ATM card allows 1,000 per day. Perhaps, you could ask to have your limit increased to cover the expenses.

You can have money sent from your banking institution to the owner of where you are staying if you have his IBAN and BIC numbers. If this owner is dealing in cash payments, you will not be the first person needing this. If he is not able or unwilling to provide this information, chances are very high he is hiding income to avoid paying taxes. Trusting him might be problematic.

In addition to money transfers from your bank, there are a number of 3rd party entities which also handle international transactions, such as Wise, XE, or CurrencyOnline.

I use Wise and OFX to transfer funds to France. Both are significantly cheaper than my bank. Wise is much faster than OFX. I have found that the French pay via wire transfer much more frequently than Americans do so your accommodations is possibly familiar with receiving wires.

As per Tocard, I have used XE for years with no problem. My experience, however is only from doing this from my home computer. Probably more difficult on the phone. If your bank card is in a different time zone than France and your limit is a per day limit You should be able to withdraw Money at 11 PM or earlier in your banks time zone and then withdraw the same amount a few minutes after midnight once again in your banks time zone. I don’t suppose you brought ATM cards for more than one account? For future reference, that would certainly help in a situation like this.

Wise would be simple, but the OP said cash or "french cheque," whatever that means-- bank check, vacation voucher such as cheques vacances ?