Europe is not that big a place, with a superb domestic airline network.
Why not just for your 4-5 day periods routinely travel *to* the flying school. Particularly if it's at or close to a reasonably major airport, this should be no issue at-all. Also you'll be learning at a known airport, in known landscape, in a consistent aeroplane. Learning in a different country each week does not sound even faintly sensible for learning progression.
Also, in my experience, people trying to juggle learning with lots of other things at once, seldom learn very well. Dedicated periods of time where you quite deliberately avoid work issues will almost certainly provide you with a far better learning experience.
I entirely understand the schools who said "nope" as they probably share my view that this is absolutely no way to learn to fly. Those who didn't reply are probably a combination of people who just thought you were winding them up, and those who (sadly too commonly in this industry) just are incapable of answering any faintly difficult questions - there are plenty who behave like that in flying training.
Also being blunt, if you are incapable of dedicating time to learning to fly, and maintaining your flying skills - you'll never be a safe pilot. Hire somebody else to fly you around Europe - if they're an instructor, and such people are easy to hire, you can keep flying under their "instruction", without actually endangering anybody.
G