ANPR cameras at the UK ports and in Police Cars will flag the car up as no MOT very quickly and there will be other problems if you have SORNed it in the UK. The UK authorities have wide powers to seize unlicenced vehicles to remove them from the highway. The lack of an MOT is a criminal offence but this can be circumvented by arranging an MOT close to the point of entry to the UK.
I would think that you might stand a better chance of proving intent if you have notified the DVLA in the UK of permanent export and can show documents (an online reciptreceipt for example) that you have indeed applied to register the car in France and if old enough, you have a French Contrôl Technique which is valid in the UK.
IIRC there are long delays in the issue of Vehicle registration documents (Certiciat de Immatriculation) in France following the introduction of a new on-line system which you might (I say might) be able to short circuit by asking a local Garagiste to do it for you. It will require the CoC which, in the case of a relatively new vehicle will form part of the UK registration document.