I think we may be done here. Fingers crossed.
I ran the pump through the 1" PVC Flex for four days. During that time there was only a slight reduction in pressure at the fountain. I expected that as I assumed that there would still be biofilm buildup; but not enough to prevent a decent flow. But the !" hose alone produced too much flow into the stream. When I showed my wife the first thing she said was, "It doesn't make a nice sound anymore." Very true.
So, I tried running the water from the 1" PVC flex through a reducer to a ½" ID piece of rigid PVC about 30cm long. This piece fits through the existing hunk of bamboo:
This is an overhead view of the elephant fountain. At the left is the blue PVC Flex. At the right is the stream with the fountain making the pleasant sound we've enjoyed for over a decade. I've been running this for four days now and have noticed no significant reduction in flow, so far. If it does happen, I'll first check to see if just reaming the rigid PVC works. If not, it will be back to the drawing board on the fountain configuration. (I left the old green garden hose in place in case I needed to suddenly revert.)
Thanks to all who offered suggestions, help, ideas and criticism. I truly appreciate how much assistance I received and how much I learned.