Anything loose in the guitar or the transducer can cause a ring or snare. This can be as ridiculous as the beginning of a string resting loosely on the top of the guitar. Or the adhesive pad (?) for the transducer might have grown loose spots. Unless the problem is in direct vicinity of the transducer, you should be able to discern it when playing loudly and listening carefully. Note that putting the guitar flat down in order to better find out things may change the problem, at worst making it go away. So try in playing position.
If it is sympathetic vibration, it will typically not be there for low volumes but will maintain much of its character and relative volume for varying large volumes. Amplifier distortion, in contrast, tends to get worse at higher volumes, not just louder.
For detecting sympathetic vibration in instruments, a cheap vibrator with rubbery surface (don't use hard surface to hard surface or the coupling point itself will be too noisy to be useful) can be used for coupling a significant amount of low frequency excitation into instruments, making it reasonably easy to detect the culprits for buzzing.
However, if you cannot hear anything akin to the buzz in loud acoustic play, any possible mechanical vibration will likely be confined to the innards of the transducer (like a loose coil or bad solder point): if that is the case, you'll be hearing a buzz from the vibrator in the amp, but not mechanically.
Apart from the mechanical possibilities, there can also be an electric mismatch between impedances from transducer and amp that leads to saturation effects. If different amps or just a preamp make the problem go away, this might be the problem here. If that is the case, a DI box matched to the transducer impedance might do the trick.
