I've seen it done plenty of times Beep... especially in the studio.
There are many ways of 'damping' the strings I've seen employed to give a duller tone richer in subs, most common way I've seen it done is by the bridge cover being packed with foam, or foam just being layered beneath the strings to kill off some resonance.
The 'problem' if you can call it a problem isn't to do with the pups, or technique at all it's in the resonance inherent in the strings, flatwounds can help but sometimes I've noticed that either the engineer or bassist has wanted an even 'flatter' tone with really shortened sustain to work with.
Sometimes I've seen it done actually where the pick is striking to keep the picked tone yet softening the attack still. Whatever gets you where you want to go I say.
Like you say, instead of having to alter the set up of the guitar pup height and stuff, it's easy enough to reverse when you want the brightness and attack back.
If there was a 'proper' way to do stuff we'd all be sounding the same you can do all sorts of stuff when nobody is looking to make stuff sound edgy, different and original then people will be asking how you did it!