Still Need Help

You cold put a hundred cinder blocks under that turntable and it would not help. Your only hope in this situation will be a subsonic filter. I doubt you will cough up for a MinusK platform. This is why suspended turntables like the Sota are so important. You can kiss this stuff goodbye. 
Any wall in the house is going to vibrate to some extent. Wall shelfs might help against footfalls but not to feedback. 
The best subsonic filters are digital as they can roll of much faster without affecting the audio band. 
It is possible that a cartridge that is less compliant with a lower output might help but I would not guarantee that. There may be a less expensive isolation platform that will work but I do not know any. Any isolation feet or platform would have to be sprung to have a resonance frequency around 2 Hz.  

Some strange opinions here. Atma-Sphere makes the most sense. No, subs and turntables do not play well together but it is imminently possible to make it work. I use a very powerful subwoofer array and I have ZERO difficulty playing the turntable at 105 dB. I use a suspended turntable resonating at 2 dB and an 80 dB/oct digital subsonic filter down 3 dB at 18 Hz. A well isolated turntable set up correctly will do tolerably well without a subsonic filter depending on how loud you go. I have not tried the situation with a table like the Rega and just a subsonic filter but, if the cartridge is chosen to give a tonearm resonance frequency of 8-10 Hz it should be OK. A turntable has to be isolated to work at the highest level of performance unless you put the turntable in another room which some people do. I actually boost the bass. Without a subsonic filter my woofers would be dancing the jig but, with the filter they are perfectly quiet until a low note comes along. Unfortunately analog filters do not work well. Yuo can't roll them off fast enough and they create issues with the bass which is why audiophiles hate them. The situation is totally different with digital filters. Yes, I digitize my turntable. It is converted t 24/192 by a Benchmark ADC 1. The transfer and back is totally invisible an opinion also relayed by Michael Fremer who uses the same digital set up I do.
Our OP needs to put his turntable in another room, get a subsonic filter or get a suspended turntable. You can suspend the Rega with several options on the market but, if the system does not have a very low resonance frequency, below 3 Hz, it will not work. The first hint that you might be on to something is isolators that are "sized" to the weight of the turntable.