I hope that these very new question pages will in time attract the attention of piano moving companies to answer and so attract clients.
But I will throw in one comment.
I can't comment on the price.
There seem to be 5 elements to consider in this.
*collection for and arrangement of shipping.
*collection and delivery from the Port of arrival.
*satisfactory crating.
*Insurance.
*Language...
Apart from reviewing and contacting a couple of qualified movers in these French pages, I would,
have a hunch it may not be crazy to contact a British mover with export experience:- They get EVERYWHERE and piano moving is all about routes, and who is going where and when... and NOT about who is based nearest.
You are short of time, but you are looking for a qualified exporter who will be in Narbonne before the 15th from anywhere! sounds tight! I would draw up a list of firms who look suitable and ask around..a lot.