I have crossed France
from the North sea to the Med in 1988 with a 32ft sailboat, drawing 1.5m. It can be done with boats having 2m keel
and even more, if you choose your path carefully. The minor canals might pose a problem but there are always alternative "highways" which commercial
vessels incl. 800 ton barges take.
In springtime the rivers are said to have strong flow - up to 8 knots - but otherwise it should not be a problem.
Since you'll have to adjust your sailing to lock-keepers working hours it will take around 4 weeks, perhaps 3 in summer.
And ofcourse, you'll have to take your mast
off, support it on deck
and get plenty of fenders and mooring
lines. I believe there are books
on the subject - mine (Waterways of Europe) is some 20 years old but new ones can be obtained. For my crossing I bought a set of "pilot books" - each about 50 pages thick, including charts
and touristic information, in addition to a single
chart covering all the waterways in France
. At the time they could be obtained in most sportshops, nowadays naturally the web should be the source.