You say your husband has a medical background, did you mean he is a doctor?
If so, this might well explain his unwillingness to let you attend hospital with him. My father was a doctor and although he latterly did accept that I accompanied him to hospital, as he was in a wheelchair, I think he would have preferred my not being there, but as he was not fluent in Danish, and not all doctors here are fluent in English, he put up with my services as interpreter.
A doctor, perhaps especially a physician, may not be able to confide his worries to his wife. I visualise a long line of doctors' wives agreeing with me here. Sorry as I am to say it, I don't think you will change him on this point.
Has he always tended to be secretive, or is this a new thing? If it is new, it could perhaps be a sign of the onset of dementia, but are there any other signs?
You may already have done this, but in your place, I would tell him exactly what you have told us, and say that you do not want this situation to continue. I would also work out exactly what all his journeys to the UK are costing , then work out a budget of your household expenses and of your joint income and put it in front of him.
As you cannot afford to go with him, you really cannot afford him to go on using money in this way, so now is the time to revise your expenditure. Otherwise, you might well find yourselves in debt.
A lot of people (not only men) become increasing selfish the older they get, and ill- health seems to accelerate their selfishness. There is little you can do about that, except point out that his attitude is making a difficult situation harder for you.
Language is a barrier when health issues arise, could you afford to pay a professional medical interpreter if he seeks treatment in France? It is perhaps a solution worth looking into.
I hope you find some solution to the problem.
I have no idea whether he is entitled to use the NHS or not, but look out for trouble when or if Brexit becomes a reality!