I am not sure why it is dishonest to continue redeeming the miles from the deceased account.

If you inherited them, then you took title to them upon death. There is no dishonesty in using what is yours whether or not someone else knows it is yours or not. The only ethical issue would be calling up the airline and misrepresenting yourself. Otherwise, redeem online to keep life simple.

OTOH, from what others have posted above, why not just make it official to be sure that problems are avoided mid-trip.