I have a similar problem with a Fishman Prefix Pro Blend. Have had intermittent hum and now slight distortion-type sound.
tldr: I taped portions of the pickup which appear rubbed off, the problem is still there and I am continuing to troubleshoot. Will try taping the whole pickup next, and if that doesn't work, replacing it (made in China).
I have tried (and read about):
Checking for anything loose inside the guitar, endpin jack washer/nut in particular, (used a 1/4 in dowel rod to carry it forward to the sound hole and check these), and adjusted to ensure the jack fit snugly. The friction washer may not be working well - as my installation had an extra one that was doing nothing but rattling around.
I also noticed a change in sound with pressure on the saddle/bridge, if this is the case it relates to the pickup and/or bridge fit - as you've noticed, tiny changes matter. I have read elsewhere the position of the pickup is important - some have said it should fit all the way to the end of the slot; others that it should never touch ends of the slot, be free to move, and the cord should not bind in its hole.
Also, the bridge should fit loosely enough to remove easily, but not wobble, and not slide lengthwise. And the slot must be free of dust or grime built up from playing.
I did notice some slight abrasion of the red foil on the pickup as though the bridge had rubbed it off, and attempted covering those areas with the two-sided foil tape to shield them and restore continuity. The sound is still there. Others have recommended shielding the entire pickup; and others the entire cable run all the way to the pre-amp. I am not sure whether to tape over the existing foil or attempt to remove the red foil first. The taping has to be completely smooth with no wrinkles to ensure even pressure distribution and balanced string output.
Another potential issue is the pickup quality itself. Fishman has already recommended replacing it for these symptoms, though it is only 3 years old and has only been played 1 1/2 years since built.
Another issue may be battery type. One person found improvement using a thinner (Energizer) battery vs a 1mm thicker (Duracell) and it made an improvement lining up with the battery compartment contacts. Others have said rechargeable batteries have lower output issues, use only disposable ones. (I'm out of batteries at the moment but trying that next!)
Additionally, power quality can be an issue. 50Hz vs 60Hz power will affect the circuit and may be why certain amps or plugging in at different locations has an effect. Power conditioners can help this issue and with power fluctuations due to wiring and other demand surges on the circuit.
There is a lot of info out there on bridge fit and especially flatness and vertical alignment; slot flatness - which has to account for arching under tension; slot smoothness; bridge height and string break angle. Anything that affects the balance of all these things can lead to issues. Including time, humidity, wear, impact, string tension, neck adjustment, action height, etc.