It can work but I think it will be incredibly difficult for your older DC.
We moved when DD was 7 and she struggled for a while. However it was easier that she was still in primary school (a good rural one by the way) as there was the same "school-gate" chat that you get in UK. This was great for making friends and also for improving French.
However once the DC go to college (often at age of ten/11) then it's very similar to the UK and there are less opportunities to integrate and chat with other parents. Depending on your 12 year old date of birth they may well be going into 6eme or 5eme - so not even starting college at first stage.
Help with assimilation - re-doubling a year - may be available, but then your DC may be in classes with kids a lot younger than them.
Don't waste money on limited French classes in the UK - total immersion is the only way and if your French is patchy don't try and teach them.
Running a BB is very patchy if you are not living in a touristy place that is open all year round. Most places (even in popular tourist areas) "close" from late September through to April. This includes restaurants and attractions that would appeal to your BB guests - there's no point going somewhere if all local facilities are closed, however attractive your BB.
Also remember you will have to pay taxes on your business - which in the early days are very, very restrictive and based on potential earnings not what you actually made.
As PP have said property is very cheap because there is often no jobs or infrastructure nearby. Nor do the French place the same value on doing-up homes - most of my French friends have or aspire to buy a new build. Please rent for a year and see if it suits you - renting can be incredibly cheap and is the best way forward (we sold up and although love France and it has been good to us - it is my biggest regret!)
Also be wary of moving to anywhere with a high percentage of ex pats (Normandy, Dordogne and Brittany) as there is the "danger" of being sucked into English-speaking activities and groups and miss out on the "real" France. You will also be competing with others who run B&Bs, market gardens, arts and craft stalls etc.. In these places there are often English-speaking kids in schools but it can restrict your DC's learning French and making their own friends.
At the end of the day only you know what's right for your family - it worked for us so it would be hypocritical to say don't do it - just make sure you do your research and have a fall-back property in UK.