Whatever the base of the small tiara is, it isn't the base of the Girls tiara. The small tiara, as illustrated by the photo of Princess Alexandra, went to the Kents, the Girls tiara and its base to Princess Elizabeth.
When the present Duchess of Kent had the small tiara remade, the base was preserved. Scroll down this link to the Royal Forums and you'll find a good picture of the Duchess wearing the new tiara, which by the way has something of the look of the Queen Victoria wedding gift tiara.
Since Roberts indicates that this small tiara was a new purchase, we could conclude that the tiara base was a standard design, used for all of these fringe and lattice type tiaras.