UK Limited company, but resident in France?

As a French resident I would like to better understand the following scenario. As a director of a UK company do I need to register the existing UK Limited business in France? Most of the consultancy work is in the UK however clients from anywhere in the world can also order product from the website. In real terms there is no business activity for us in France and even then, payment transactions and order fulfilment would be from the UK. As a director this company (resident in France) there would be an element of teleworking from my home in France together with the commute to the UK. At this stage this commuter activity could be around 90 days of each year.

With the Limited company is it correct to say that coporation tax / PAYE can continue to go to the UK Authorities as per usual, and I declare my worldwide income on the French tax form? How does that work if you pay yourself a small UK salary and live in France? How can I best dovetail into the French system in order to make social contributions here etc without setting up an expensive SARL / EURL? So, whilst not looking to expand locally at present, could it be possible to say set up as a registered commercial agent of the company in France under one the micro-social schemes..that way can start to make some contributions here?