What kind of cover?
Insuring your bike against theft or loss (not while riding)? Usually house insurance *should* cover it away from home so long as in your possession or secured properly. Maybe premium for outside UK.
Insuring bike in use, then need MTB specific policies usually although some standard travel insurances might cover it but often there are many exclusions and they might only cover MTB so long as it’s on essentially tarmac or flat gravel fireroad type stuff.
Likewise with medical cover and again travel insurances may limit it with riding off road and also watch out for any exclusions for “out of bounds”, i.e. off piste, like with skiing.
Remember EHIC card, but note that this just gives you basic medical attention in a state hospital, probably not mountain recovery. Travel insurance required on top.
Racing? Need specialist insurance (and don’t forget medical certificate for France else may not let you race).
Anyway, suggestions… Snowcard and Dogtag.