Realistically, there's no way to do this without a major military component, which would be yet another "go ahead, give me a reason" action by France. There's not much legitimate room for Russia to protest,
Russia will accuse France of using "humanitarian aid" as an excuse to bring its own troops onto foreign soil for the sake of interfering with Putin's "Special Military Operation", while utterly denying the irony of that statement.
Putin's already an international Pariah. His only goals are to keep his citizens duped and to avoid having his soldiers directly shoot NATO troops (cause that would lead to nuclear war). When Putin says no, France might send in the aid anyway, and at that point, he'll have to decide whether to shoot the aid. Maybe he'll just bomb them "accidentally" since they did walk into an active war zone.
That won't actually work though. Putin's lies are only meant to convince the Russian people. NATO will see the attack for what it is, and then Russia will be fucked (along with the rest of us probably). If they don't shoot the aid though, then Putin will look like a weak idiot when the fearmongering in this hypothetical initial objection doesn't pan out.
Hmm, maybe it can work out then. Just depends on whether France, Turkey, and Greece has the balls to actually send it in if Russia says no. If Russia says no and they actually respect Russia's request, then they just gave the enemy a win.