I understand that many, if not most currently available pickup systems (systems includes the on-board preamp) exhibit compression...it's called clipping. It's not the pickup that's doing it, it's the underpowered buffer preamp stage. You are hearing the compression of a buffer clipping, not of the pickup. It's very important to know how to isolate these issues in order to be able to solve them.
Also, I'd be very careful about calling the effect "compression"; it's really more like peak limiting...with distortion.
In a test done by dropping a 1/2" ball bearing 12 inches directly onto a piezo film UST on a table, we have measured in excess of 100 Volts as the initial transient peak. That's with a 100 meg Ohm probe directly into an oscilloscope. No compression. I invite you to duplicate that experiment.
So OK, you're not going to get peaks like that with a pickup installed under a saddle, but you may very well get peaks well in excess of 9 Volts. That spike may only happen on the very first half cycle of string motion, but what happens to a 3 or 9 volt buffer preamp when that happens? It clips, and it then takes a little bit of time to recover. That phenomenon is a major portion of what's often referred to as "piezo quack"...it's the very first stage of preamplification clipping the peak and then recovering...politely or not so politely.
We are used to longer clipping events than this 1/2 cycle phenomenon because it's usually interstage in a chain of electronics, and it's usually sort of symmetrical. It's not with UST pickups because of how incredibly strong the first half wave of string motion is.
I have been an advocate of higher voltage on-board electronics since my Alembic days in the early 1970s. You can't go wrong with giving the pickup a good interface, and you can always pad down a signal that is clean and not clipped, but once the signal is clipped, you can never get it back.
When we (D-TAR) came out with our Timberline and Wavelength pickups, the universal comment was that we'd tamed piezo quack. There is a component of "piezo sound" which is not the technology, it's the location. Note that the B-Band pickups...which use an electret foam element...still sound like under-saddle pickups. It's because they're under-saddle pickups! You're hearing location. It's a very phase coherent sound...unlike the true acoustic output of an instrument which takes the string vibration signal and imposes all kinds of phase delays at different frequencies.
Some of the Wavelength systems...the MultiSource models...do have a mic with the lows rolled off below 200 Hz, but that's not what I put in ukes...it's just not necessary. I use the Timberline with an 18 Volt power supply, and it sounds like a nice uke.
I have put decades into learning the "secrets" of this stuff...the theory, the practice, and the limitations of many kinds of transducer designs. Oddly enough, some of my understanding of how acoustic instruments really work (as in change the vibration signature of a string into acoustic output) came from helping to design a digital acoustic instrument modeling preamp. I suffered through many epiphanies in that project which have ultimately helped me with the design of true acoustic instruments as well as in understanding transducers better.
BTW, the term "transducer" is too often limited in our world to contact pickups...soundboard transducers. Technically, a transducer is anything that transforms physical vibration into an electrical signal...or vice versa. Magnetic pickups, microphones, USTs, SBTs, loudspeakers, headphones...they're all transducers.
