Can tourists get French cheque books?

Pre-paid cheques do exist, but they're not that widely used today. The most common form of them is Travellers Cheques, but they also exist as Bank Drafts, Postal Orders and a few other things. They have a few drawbacks, including needing to be sorted out in advance, only coming in pre-determined amounts, and either needing the recipient to known + named at issue, or otherwise security risks if anyone can use them.

I'm not sure therefore that a pre-paid cheque would be all that great a fit for your use case, since you wouldn't know before leaving Belgium who you'd want to make a cheque out to a few weeks later, nor exactly how much for. Travellers cheques in conjunction with some cash (to hit the exact amount needed) might work, but they normally attract a fee to buy, and not all places will be comfortable accepting them and cashing them in later.

That all said, there is one kind of "pre-paid cheque" which is incredibly popular and widely used in France - Chèque Déjeuner (and similar, often simply called ticket resto). These are normally purchased in a large number by a company for its employees, and then exchanged for a meal (typically lunch, but not exclusively). As they come with some tax breaks, they won't normally be available to you as an individual on your travels. They also won't help for the specific case in your question, no matter how widely this specific kind of pre-paid cheque is used in France!

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So, with pre-paid cheques largely out of the running, what can you do? As I understand it, you need a euro denominated cheque book. Nothing about it needing to be a French one, as long as it's in euros and from a European bank (or at least with routing details via one). My first suggestion would be to ask all the banks you deal with at home, and see if they'll sort one out for you, most likely for a fee.

Otherwise, you'll need to find a European bank in a country that still does cheques, open a Euro account with them, transfer some money over (luckily SEPA means there'll be no charge for that), and write the cheque from your other account. It doesn't even have to be in a Euro country - for example Citibank London offer Euro accounts and these come with a euro cheque book!

I'd suggest you start by asking one of the more international banks in your home country, and see if they do offer cheque books - they may do if they're used to their customers travelling a lot more. Next, check if one of the international banks elsewhere in Europe will give you one as a non-resident, and compare their fees. (The Citibank London one I mentioned can be free depending on how you use it, or otherwise has a monthly fee, so decide if it's worth it). Failing that, you'll probably need to open a French bank account as a non-resident. As for how to do that, a new question on Money.SE would be your best bet.