Why the woofer moves badly when playing certain LPs

The idea of going to lower mass mounting screws is a good one (worth a try- its cheap; maybe also remove the stylus guard), better yet a cartridge with slightly lower compliance.

A cartridge with lower compliance (15cu) that i mentioned also has light mass itself (5.5 - 6.5g instead of 8g AT). Could be an ideal solution if the OP's problem is cart/tonearm resonance.

@starbusters look at Pickering 3000/SP:

Stylus Type: Nude Stereohedron
ORIGINAL STYLUS: D3000SP
Compliance: 15cu @10hz
Contact Radii: .0028 (71u)
Scanning Radii: .0003 (8u)
Stylus Tracking Force: 1 gram (+1/2g) (-1/4g)
Setting with Brush: 2 gram resulting operation tracking force 1 gram. Optional range is (+1/2g) (-1/4g)
Frequency Response: 10 Hz to 30 kHz +
Output: 5.0 mV
Channel Balance: Within 1 dB @ 1kHz
Channel Separation: 35 dB @ 1kHz
Cartridge DC Resistance: 600 ohms
Cartridge Inductance: 350 mH
Cartridge Color: Brown
Cartridge weight: 5.5g (6.5g with brush)
Load Resistance: 47-100k Ohms
Load Capacitance: 275 pF
SP version was made in 1989, discontinued in 1994

Cleeds, since when is a resonance point a "brick wall" situation. The resonance peaks then rolls off. If your woofers do not move at all visibly and you are playing vinyl either your woofers do not go down very low or your cartridge is too stiff for your tonearm and your bass is rolling off prematurely. I suppose you could also be listening at the volume of a mouse squeak. I would never use an analog filter. It would damage the bass too much but digital is a whole other world. I could survive without it but volumes with certain records would have to be limited and It might rob enough power to affect the room control function. There are no downsides to a digital filter up to 80 dB/oct. With a 3 dB down point at 18 Hz the effect is totally inaudible. 
The problem Cleeds is that the source is vinyl, a medium that is imperfect at best. So, treating it at it's source means not playing records.

You need a 6/9 dB per octave (the higher the better) subsonic filter at around 18Hz.  Very unlikely you will hear anything below 25Hz anyway (more \like "feel it" but that will take care of it.

I see a periodic pattern in the woofer oscillations: every so often there’s a slight pause or dip in the amplitude of the oscillations and then the pattern repeats. The video doesn’t allow a precise clocking, but the pattern repeats roughly every 2 seconds. Given the observed warp in the record, I’m going to speculate that the pattern repeats 33 1/3 times per minute. Rather than an instability at the tonearm/cartridge resonant frequency, my guess is that the warp is triggering a tracking error in which the cartridge "bounces" along the groove without loosing contact altogether. This would imply suboptimal reproduction at all frequencies, not just the observed woofer misbehavior.

If this theory is correct, a rumble filter or adjusting the tonearm/cartridge resonant frequency would not correct the tracking problem (although I expect the latter would alter the frequency of the bounces). A first test of this theory would be to observe the woofer and count the number of pattern repetitions in one minute (should be about 33). A further test would be to increase tracking force and see if that reduces the amplitude of the woofer’s oscillations (might not eliminate them). Beyond tracking force adjustment, you might try adding a peripheral ring record weight. That should significantly flatten this relatively modest warp and add some extra flywheel stability to your rig.

BTW: Adjusting tonearm/cartridge resonance is worthwhile in any case.

If this theory is correct, a rumble filter or adjusting the tonearm/cartridge resonant frequency would not correct the tracking problem (although I expect the latter would alter the frequency of the bounces).

This statement is incorrect. If the mechanical resonance is corrected the amplitude of the woofer movement will be decreased. Making the LP flatter would certainly help, but is a more expensive solution (although it probably would sound better for other reasons such as improved control of the resonance in the LP itself) than simply reducing the mass at the end of the tonearm.