When we bought our current LHD Renault Espace in the UK 8 years ago and on UK plates, I checked with UK insurers and several would not insure an LHD vehicle. Most car insurers have a rule that any modifications must be notified to them and even small modifications that have no bearing on the vehicle safety or performance can be taken to have invalidated the insurance. Caveat Emptor is often the case with insurance - I accept a vehicle constructed as RHD or LHD cannot be considered to be modified from a logical perspective but legally it may not be so simple - legal matters so rarely are.
Direct online insurers are a potential minefield. Their business model is built on making things simple and “efficient” but that comes at a cost, which is that they focus on non-specialist cases in order to make the bulk business that they depend upon for that model. In many insurers, non-standard cases are judged separately and premiums adjusted accordingly where they accept the risk, but not these direct insurers who work mainly to computer algorithms. When I took out my UK insurance LHD or RHD was not mentioned on comparison sites and it was only later when I checked that I discovered this exclusion. (The car is since re-registered and insured here and of course without that problem).
And, actually, let’s be honest, there is an increased risk involved in driving an LHD vehicle in the UK or an RHD vehicle in France. Whether you choose to overtake on a single carriageway road ahead or not, your view of the road is more restricted.
Again legally, most insurers would have difficulty wriggling out of paying third party claims following an accident if they had been fully informed of the risk and accepted it anyway, even if their acceptance was based on their own negligence, but fully comprehensive cover might still be forfeited. In theory, therefore, you might have a valid claim for retrospective refund on the difference between third party and comprehensive insurance (which is often not a huge amount) but if it was contested I wouldn’t be confident of winning. Some companies will make a goodwill offer whilst maintaining they are not liable and that is the best hope IMHO.