Thanks Dr Tom.Ah hah! You don't have a Douay-Rheims bible. You have a Challoner bible.
Richard Challoner published three editions, 1749, 1750, and 1752. The Catholic Church retained the title Douay–Rheims Bible, but the Challoner revision was a new version, which took as its base text the King James Bible.
So, it is not the KJV which uses a Rheims reading, it is the Challoner/Rheims version that uses a KJV reading.
HankD