I have a Deluxe Active Jazz and a Reggie Hamilton Standard Jazz, both MIM, and both pick up noise when I play them at church and am not touching the strings. I would not call the noise hum because we generally reserve the term hum for 60 Hz noise which is a low bass note (between the B and Bb on your E string) and the noise I get is higher pitched than that. I suspect yours is similar since you say it is affected most by the treble control on the preamp, as is the noise that I get. I would not describe the noise as very loud but it is easily audible and of course a church is much quieter than most concert venues would be so even a quiet noise can be a distraction. For the time being I deal with it by keeping my hands on the strings when I am holding the bass and by turning the volume all the way down when I am not.
You say your bass is shielded and when you say that I assume that you mean it has the same factory shield paint job that my two have. The DAJ has a fairly good shield paint job, whoever did the shielding on my RH was clearly having a bad day. Some day I mean to put in proper metal foil shielding on my basses. This may or may not kill the noise pickup. People often report that it does yet some who have done an excellent job of shielding still can have noise problems. Some interference sources are stronger than others and you cannot put a complete EM shield around your pickups without making them unable to sense the strings so the door is always open to a strong enough interference source.
There do seem to be fairly consistent reports of noise with Fender preamps. The MIA preamps seem to get the most complaints but maybe that is simply because those who pay more will complain the most about problems that may be common to all active Fenders. So it is possible that Fender preamps are more susceptible to noise pick up than others. You say that a bass with active EMG pickups does not have the same noise so there obviously is a fix for the problem. I don't know what the issue might be, perhaps the EMGs have good internal shielding. If so a better shield job on your Fender should fix the problem. Otherwise it is possible that Fender preamps may need some better RF bypassing or something like that. Unfortunately it would take a lot of time and work to sleuth that out. I guess I should take my digital oscilloscope/spectrum analyzer to church some time and see what I can tell about the problem by looking at the noise spectrum....