I don't see that operating an AE (or a traditional Micro or other entreprise individuel) would be a problem alongside UK employment, per se. You declare the approriate amounts in the two countries and pay social security (NI / cotisations) on each. As I understand it you'd be defined as a French residence by having a French address and making a tax return. Of course, as you know only too well, that doesn't stop you being a UK resident (with its variations) too.
I don't claim to be 'properly qualified', but if you excl;ude HMRC and the French tax office as well then it starts getting difficult to identify anyone who could advise.
Albert the InfoGipsy
"So welcome to the Citadel where the question is 'Am I?'"