Trying to get the best out of cheapo piezos

In former times I used to dampen the back side and even putting a second layer of alu foil on the backside.

You have to unsolder the wiring for that gently putting the paper cone up/out without deforming it.
This is true for this style piezo. But for the larger style ( e.g. https://www.parts-express.com/GRS-P...-Tweeter-Similar-to-KSN102-292-444?quantity=1 ) they come apart more completely. Unscrew, and the horn / phase plug comes off as one piece, and then the piezo element + diaphragm come off, and the back comes off as a cap. More places to potentially leak air, but allows easy access to the back of the piezo and diaphragm. I considered unsoldering mine to reach the back but figured I would melt the plastic housing. I've already picked up some of these larger ones to experiment with. In addition to potentially being more leaky, they also seem to have more molding flash on the plastic parts. But more friendly to mods, and the plastic is thicker.
Even adding a rubber damper to the Piezo like in the early Motorola piezos is worth a try.
I see the foam and rubber dampers on the diagram in post 2. If anyone has pictures or more detail about these parts I'd love to know about it. Apart from dryer lint ;) which actually seems like a really good density for this sort of thing. I've been considering what sort of material (and how to attach) to use for the back side of the piezo element.
What I like most with piezos that if filtered properly at least with minimal crossover of 6db they lose their nervous sound everyone experiences with the raw drivers.
Yes, I think unfiltered the LF signal doesn't make sound but still uses up the deformation range of the piezo element.
 
So perhaps it is core saturation which is the limiting factor?
Yes, but it depends on the inductance. For the same core: higher inductance = lover rated amperage (saturation)

I wonder if there is a slightly larger ferrite core version available...
You don't need them for a tweeter filter - they are more expensive. If you are willing to pay more, air core inductor is much better.
With the same small ferrite core, rated amperage is between 0.77 and 0.85 A for inductances between 0.22 mH and 0.33 mH.

0.33 mH, 850 mA, $0.68
https://www.digikey.com/en/products/detail/bourns-inc/RLB0913-331K/6677227

0.27 mH, 770 mA, $0.42
https://www.digikey.com/en/products/detail/bourns-inc/RLB0914-271KL/2352778

0.22 mH, 800 mA, $0.42
https://www.digikey.com/en/products/detail/bourns-inc/RLB0914-221KL/2352777
 
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