The rules don't actually say 'the
original Amex card' - although it could be implied.
All Taxes, Fees and Charges must be paid for using the British Airways American Express Card.
We cancelled and re-booked a 241 trip (pre-Covid) after the original card had been cancelled. Everything was refunded with no problem, although we had to get Amex to shift the refund from the closed card to our current one. The re-booking was done with the original voucher and fees were paid from the new card without any issues.
BTW, if you intend to 'downgrade' to the free card you don't have wait until the renewal date - you can do it any time and with Amex you get a pro-rata refund of the card fee.
Editted to add: You can also cancel the trip yourself if you wish, although you would lose £35pp as a cancellation fee. Clearly, if you wait until the flights are cancelled you would get it all back - but they might not be cancelled. But - depending on when (or if) you want to re-book, the sooner you get the points and 241 voucher back, the sooner you can re-book, with maybe more chance of availability.