This is mentioned as being difficult at lunchtimes, bars not serving food and bakers shutting at 11-30 'til 16-00. Whilst I can't vouch for the whole of France this is mostly incorrect. The bakers will normally be open until 13-30 and reopen at 16-00.
A bar will often be just that, no food served at any time. There are many small family bar/ restaurants that serve food at lunchtime. The thing to seek out a place advertising 'Menu Ouvrier', this is the 'working class lunch' . You can get an excellent meal (3 courses) for 12- 14 €. Often these places look very humble from the outside but don't be put off.
For me you can keep the fancy priced restaurants, give me one of these small places every time, nourishing food, just what you need.
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A cyclist unused to France would be well advised to have a backup plan if hoping to buy food and water over the counter for a large chunk of the afternoon as it's not unusual for shops to be shut for a prolonged period in the day. I was caught out without water. I dare say a local may top up from a fountain and ignore the "do not drink" signs................
Good tip about Menu Ouvrier. I have used these in the past but didn't know they had a name.
About opening hours, I think a lot of foreign visitors tend to have a distorted view on this based on reading old comments or books.
Small shops that run without external staff (bakeries, small independent grocery stores or "mini markets", etc.) will have to close during afternoon hours because they open early. From 12h30 or 13:00 to 15:30 or 16:00 usually. Supermarkets tend to be about anywhere a larger community resides and they will be open 9 to 19 or 19h30.
Nowadays anyone can check out on Google maps to see examples of opening hours.
About the menu ouvrier, it's a good tip and quite true they are very good value, but unfortunately places that advertise it as such are few.
Best way to find them is to spot restaurants that don't look too fancy and even better, spot workers' utility vehicles parked nearby.
If you run short of water and bars and shops are closed you might be lucky to pass a cemetery. I've always found a tap somewhere in them, not always obvious. The water will come from the mains so is drinkable.