I'd try the foam to dampen, but maybe keep the springs too - Warwick use a nice threaded insert and it would be a shame to lose the positive support. My guess is if its that noticeable its going to bug you, so needs sorting out.
Make sure the treble isn't boosted - easy to do with a new circuit. It could be a dry joint - these can respond to physical vibrations, but the finger noise does sound like microphonics within the pickup casing to me. Have a coffee, double check the whole circuit again and see if anything else makes noise when it gets waggled. Also turn the preamp gain up to half way - it may be acting strangely at its lowest setting.
The only real test is to hot wire a pickup to the jack, and see if it makes microphonic noises?