That sounds like quite an adventure!
I think you mean St James Way from Reading to Southampton, though you can divert at Winchester towards Portsmouth and the cross channel ferries.
As for the route across France, I suggest you find yourself a copy of the 'Lightfoot Guide to the Plantagenets Way'. Assuming you cross the channel to St Malo or Cherbourg, there are good footpaths/cycleways to Mont St Michel. The Plantagenets Way links Mont St Michel to the Voie de Tours, at St Jean d'Angely. It's arguably more scenic and certainly more direct than the coast or going via Paris. Though it is not properly signed on the ground, it links well maintained paths and lanes and is easy to follow from the book. From St Jean d'Angely, the Voie de Tours is a reasonably well marked route to SJPdP.
There certainly will be opportunities for wild camping along the quieter stretches of the Plantagenets, assuming you are discreet and only there from dusk to dawn, but do watch out for signs relating to hunts (La Chasse); not ideal to wake up in the half light among trigger happy hunters!
Staying in a campsite every few days would help you charge your phone and get a shower. Many of the smaller French campsites (and indeed some of the more commercial ones) offer a special cheap hiker's tariff, so it needn't be expensive. Once you get onto the Voie de Tours, there are Haltes Jacquaires and Gites d'Etapes which offer basic accommodation at remarkably low prices, designed for people doing just what you plan! So you needn't end up roughing it every night. If you look up the threads on this Forum marked Voie de Tours you'll pick up some more useful info.
From St Jean Pied de Port, you need to think whether you want to carry your tent anymore, or to post it home. Suitable places for wild camping along the Camino in Spain are rare, though some people do pitch in the gardens of albergues.
Happy to help answer any questions you may have if you DM me through this forum.
Good luck and buen Camino!
