Just to make sure I understand what you are trying to do;
You want to fly yourself from somewhere in England to somewhere in northern France – a distance of about 400nm (So say Carlisle to Bernay?) – a better idea of the locations would help with the advice!
You want to do it once a month – or more.
You have a budget of 20k for the aircraft and 10k per year maintenance (max 15k)
Your instructor has advised against a 2 seater - wouldn't make for a comfortable regular ride
You would like to take the wife and children up, eventually.
First point I would make is that I fly a 2 seater all over and do not have a ride problem. This may expand your choice as very few aircraft with 4 seats are 4 seaters!
What you want to do is entirely possible provided you enjoy flying sufficiently to do it. You are talking a 4 hour flight, which is most peoples limit. I would rather fly two legs – but that will take longer. In summer you should be able to make the trip 2 or 3 weekends a month, in winter probably once every 2 months. In the height of summer you could easily fly down on a Friday evening without much bother. A PA28-140 would probably do – it has 4 seats but is at best a 2+2. There are some nice ones about for 18k ish – you would probably hit the 15k in maint in the first year, but less than 10k average after that. Increasing your budget to 30k and assuming 10k less in maint over 5 years would greatly expand your options.
Rod1