If I had received such a letter I would contact the bank.
But also check details in the letter, such as logo, addresses, phone numbers, legal statements on the bottom and compare with bone fide communications you have received from you bank. Scammers often do make slight mistakes either in the details, spelling, syntax.
Since the Government will already have details of your earnings, any expenses against earnings and bank accounts and investments via your tax return... and French banks provide the Impôts with such information if in France; for foreign accounts and investments (pension funds too now) you must declare them on your tax form... it raises the question why your bank is asking for this information since it is unlikely to be for the Government as the Government would already have this.
I do know that in the event of making large cash transfers, the bank is obliged to report this above a certain value. I recently bought a property and the Notaire payment instructions said that if paid by bank transfer, my bank had to confirm the origins of the funds... this apparently did not apply to payment by cheque... how exactly my bank did this I do not know as they certainly did not contact me about it before making the payment.
Normally details of earnings and outgoings may be required by a bank considering a loan/overdraft.
If as the letter states this has something to do with money laundering, it seems unlikely it would be something peculiar to Barclays, every bank would be sending out these letters and everyone would be getting one.
Either your bank is 'doing its own thing' or it is the first and we all may expect such letters in due course. I think if this were happening large scale there would be reports in the media because it would make a lot of people unhappy.
My instinct is this is a scam designed to get your personal information: it is easy enough these days to produce stationary with the correct letter-heads and return addresses and since banks and businesses use PO Boxes or a variety of addresses, easy enough to have reply envelopes addressed to the scammers' PO Box/address not the bank's and scammers can be vey convincing - they need only a 2% to 4% success rate to be quids in.
Best therefore to check with the bank's fraud department. You could even call your bank HQ in the UK to see if they can shed some light... whether it is some corporate policy, something specific to France, or a con.
Bonne chance.