Surely then if a French player is a member of a French club, they would be treated as if they are a member of their Federation? Is it really that complicated or am I missing something?
You aren't, but this goes back to a piece of misrepresentation by the ECF and BCF before it. The misrepresentation was that the FIDE rules on rating meant that the ECF could no longer finance itself by applying a levy to organisers and that individual membership was needed for participants in FIDE rated events. Furthermore when the ECF was founded, the word member was interpreted to mean member in the sense of member of the company, in other words a shareholder or in the ECF's context a Guarantor.
It was Sean, who as a Congress Organiser, did much to chop away at the misrepresentation by noting that whilst he could check the membership status of English players, was he really expected to check the status of players from all round the world, some from Federations who didn't run personal membership schemes? Under FIDE's draft proposal, organisers still won't have to do this, but they will be required to ensure that all players are listed on the FIDE website under a national federation, or at a cost under FIDE itself before sending in the rating reports.
You can look up a player as FRA on the FIDE website. It doesn't tell you whether they have a French licence unless you go to the FFE site as well.