I finally got around to this now and I solved my issue using thick grease as suggested by spiceBar, while also doing a full silent mod at the same time, as described
here. First let me say this: the biggest concern I had with regard to doing it was actually moot. I have to say this loud: YOU DON'T NEED TO BREAK THE SWITCH HOUSING TO GET IT OUT OF THE PLATE. Okay, now that I've got this out of the way. Let's me give you some details on my process.
The hardest part of the whole process was actually opening the case. I used a screwdriver and a thin plastic knife-like tool (similar to what you can see in the video linked in OP). However, instead of starting at the front of the case, I started at the back of the case, and I unflipped the two stands, so that I could use them to pull the bottom part of the case out, as there is no other other good edge/lip to grab onto.
Once I got the plate separated from the rest, I poked around the switch housings. My initial theory proved correct, the switch that was making the loudest noise had the most play against the board. In fact, I then realised that the majority of switches had some degree of play (even the ones that felt stable with the board assembled), there were only about 10 that felt completely solid. After seeing this, I now completely understand why the designers of HHKB decided to merge the housings with the plate. They were clearly unhappy with the amount of tolerance on the Topre switches, and since the switches are not soldered, the soldering could not be an avenue for improvement, so they just went ahead and merged the switches. I can imagine a tiny stem and solder would work just as well, though.
I proceeded to take the loudest switch out first, thinking, if I break something might as well break this one. However, it turned out I was able to take the housing out without breaking the plastic around the clip, and quite easily too. I am not sure exactly how the person in the video was doing it, so it was either my process, or some potential difference in material (his keyboard doesn't look like the new dye-sub version). I rested my left thumb on the top of the switch, pushing down gently, while using the tip of a flat screwdriver to push horizontally into the clip, as pictured below.
The first switched popped out quite easily, so I proceeded to remove more of them. While doing this my OCD kicked in and my tolerance was getting lower and lower, so I ended up removing every switch that had the smallest resemblance of play. There was only about 10 switches left on the plate by the end
I applied EK blue high viscosity grease around the hole of every switch I removed. After replacing the switches, most of them lost all play they had before, and the ones that still had play stopped producing any noise while doing so. I was really happy with how well this worked.
As an extra tip I will provide some info on the silent mod portion of my work. I mostly followed spiceBar's method, while combining in the ideas from intelli78. To crush the landing pads, I used iron and placed the pads in between two sheets of paper on a hard surface (my kitchen bench). I played around with the number of pads I could smash at the same time, but found out I couldn't do it any more efficiently one by one than I could in larger batches. So I settled on doing 6 of them at a time, with heat set between medium and high, pushing down for 30 seconds with my whole weight + arm strength 3 times (flipping the pads to the other side in between). Here's a picture of the result. Not sure if I could have done anything different to get them even thinner.
When putting the pad onto the switch, I had to stretch the hole to fit around the thicker stem, which caused the sides of the pad to flip downwards. The stretching introduced an extra complication: you can see a little build up of material right around the stem when pushed all the way down. If I tried to flatten that material any more than you can see on the picture, the rest of the pad would swap the orientation and flip upwards, thus rendering the switch unusable. Obviously, I could've used glue to make sure the pad stays flat against the switch, but the whole idea of this mod is to be non-destructive, so you can revert it or change/improve it later if you want. So I settled on leaving the pads more or less the way you see on the picture. Luckily, if you look closely at the switch housing, you can see the slider hole does not transition sharply from the flat housing, but has a little bevel/curve where this extra material can fit, before the rest of the pad hits the top of the housing, so I don't think this issue has any additional impact on the feel of the key.
I used intelli78's idea of using a strip of 3 landing pads on the large stabilized keys. I first applied it as-is, then inserted the slider into the housing to see where it collides, and then trimmed the landing pads accordingly.
The final result: the mod is as awesome as anyone else would tell you
The keyboard is now very silent and feels super solid. I wish I could more accurately recall the feel of stock silent Realforce when I tried it in Japan, but that memory is now too distant to make a fair comparison. One thing is certain though: after silencing all the extra rattle, I now clearly feel that a 45g topre doesn't quite have the tactility that I would like, and I just keep thinking about how much I want to silence my RF 55g. However, I have different plans for that - buying the silent version and transplanting the plate when I go to Japan again (soon).
