Not being resident in France I don’t have any experience with the French system so do take this with a pinch of salt as far as the specifics in France go. The following comments are based more on UK experience.
There are several approaches to VV’s - I would expect a good surgeon to discuss the merits of them with you. Off the top of my head there is traditional stripping, sclerotherapy, radio-frequency ablation and laser ablation. As to the anaesthetic - I would expect that the final say would be by an anaesthetist, after all it is their job to put you to sleep (and, more importantly, wake you up again) for the operation. In the UK you would be referred to an anaesthetic assessment clinic - that said if it is very obvious that any co-existing medical problems would make the risk high I would expect you to be offered an epidural for a VV strip.
Obviously practices vary but I checked a couple of French websites - not sure how reliable - but epidural came up on more than one so I would imagine it is not unheard of.
On a more general note I am quite surprised that the TC is so restrictive that a procedure cannot be funded socially if there is a genuine medical need.
400€ is actually very low for laser treatment, private fees in the UK are around 5x that amount. If you can afford it it might, indeed, be best to go ahead with the laser treatment. Of course if you can’t then 400€ might as well be 4000, or 4 million in which case the only option seems to be to talk to the mutuelle and talk to the surgeon and see if something can be done which will be funded.