Hello Altselgreen and welcome to the forum
A Notaire is there to make sure the law is complied with so acts for the State. If you use a French notaire of your own, the fee is split between the two of them.
Regarding using your own UK solicitor, I think a lot depends on the value of the property you are buying, whether you have neighbours, whether there are likely to be problems regarding boundaries - it's absolutely vital to check these. You'd be surprised how many people buy properties and then get a surprise when the boundaries/parcels of land aren't quite what they thought !
I imagine you are anxious not to forget to ask some vital question - something a solicitor experienced in French conveyancing may be able to help you with. I don't know anyone who has gone down this route so cannot offer you any feedback on this.