Why the woofer moves badly when playing certain LPs

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.

I am not a big fan of periphery clamps. It is a lot of extra weight on your platter's bearing and they are a PITA to use. Vacuum or reflex clamping are better solutions. Vacuum will remove some of what we are seeing but not all of it. Starbusters, get one of these  https://www.musicdirect.com/analog-accessories/hifi-news-test-lp-producers-cut
and tell us what your resonance frequencies are. This could be a great learning experience. If there is a miss match here it will be fun to see how correcting it changes your woofer flap.