Can a French registered business have only one client?

I am currently registered as self employed in the UK, and do occasional online work for a UK organisation, for which I invoice every few months. I complete tax returns in the UK on this basis. I am thinking about moving to France full time to renovate a house (for future gite rental) but would like to continue to do the online business work in France for the UK organisation when it is available. For the time being this would be my only form of employment and income, and they would be my only client although I would like to find similar online work with other companies if possible.

1. Is there a way I can continue to work for the UK organisation, doing work as it is made available, but move to France, do the work from France, and enter and pay towards the French social security so that I can pay my taxes in France and access the basic health care system?
2. I thought I would be able to set up as a micro entrepreneur for this but having read some of the forum questions and answers, I am now not so sure it will be possible. The fact that they are currently my only client would be an issue? The work is solely online, I never go to their offices, and I follow my own schedule/hours.
3. A potential alternative is for me to go back to the UK for a month or two, and physically do the work from there in a block, then return to France. Would this make any difference to how the income is treated? Resident in France but the work is undertaken in the UK for a UK company?

My main concerns about the decision to move to France are:

• That I am able to continue working for the UK organisation as and when the work is available.
• That I am able to access the basic French health system as I cannot afford full private health insurance and I believe I have to be covered by something whilst living in France.