As you noted this cannot be part of Queen Alexandra's copy of the Regal Circlet. If the Fleurs de lys come from any existing jewel, it would have to be Queen Victoria's Regal circlet. We know that all the upper elements could be dismantled.
Queen Victoria wearing Regal circlet
Queen Mary photographed wearing Queen Victoria's Regal circlet.
Nevertheless, I feel that the design and proportion of the Fleurs de lys in the painting in Queen Alexandra's family order do not match those in Queen Victoria's regal circlet.
Queen Alexandra's family order
But, as the crosses and the fleurs de lys could be dismantled, it is possible that Queen Alexandra improvised and wore the latter on a simple collet diamond base. Or perhaps what we see is the result of the artist using artistic licence!