So I realize after reading the posts that the Skerton needs to be stabilized essentially at two points, at the top, and one as far down the steel shaft as possible without interfering with the beans going down the burr. I implemented some of the advice on this thread with some added modifications as I live in Shanghai now, and do not have access to drilling equipment or nylon spacers/screw insulators. Hope this might help others who are in the same position as me.
Disclaimer: I do not have a camera, so I am reusing some of the images in this thread and using them to better illustrate what I have done. Also, while my method works, it makes regular cleaning of the Skerton a bit of a chore, because reassembling everything takes around 15-30 minutes.
Equipment Used:Plastic Tape (2mm)
Pen, Paper, Ruler
Scissors
7.5 mm drill piece
Semi-hard transparent/translucent plastic cover with enough flat space
Gloves
Top Stabilization:
Plastic Tape instead of Nylon Spacers/Screw Insulators
First, disassemble the Skerton completely, including removing the four screws at the bottom of the burr assembly. Since I cannot find nylon spacers/screw insulators in the local equivalent of Home Depot here in Shanghai, I used plastic tape (2 mm width) and wrap it around where the nylon spacer would be beginning just below the last thread, and just wrap until you get it to fit inside the tower.

Now you have just created a "plastic tape spacer." It will be a tight fit at first and the shaft may have a hard time rotating, but once you rotate the shaft a few times, the tape will start to compress down a bit, and the fit will be perfect.
Once you do this, you will notice that there is adhesive on the top and bottom sides of the "plastic tape spacer" which will cause coffee grinds to adhere to it. To prevent that, and to make cleaning easier, I would suggest that once you get down to the right fit, remove two full rotation of the 2mm tape, and then wrap a larger tape around the area, so every time you clean, a simple wipe would do the trick or just replace the outer larger tape.
Bottom Stabilization : Adjustable Plastic Plate with Hole and Slit
Now that you have the top stabilized, the bottom will still need to be stabilized as I found out unfortunately. This means more work!
I did a modification of the metal and plastic plate suggested in this thread by EricBNC and AD. I came across two problems with both methods as I experimented with them using a plastic plate.
First, no matter how hard I try to create the hole in the middle of the plastic plate so it sits perfectly centered, when I assemble everything back together and try to grind, the inner burr rubs against the outer burr on one side. This means that either the shaft is not as centered as it should be, or the center hole on the plastic plate is not centered once assembled, or my plate fitting into the four screws messed up the centering, or the outer ceramic burr is off centered. As a result, while the grinds are slightly more uniformed than without the plastic tape, there is too much coffee bean dust that clogs up the filter and creates an overly bitter coffee taste. The workaround is to make the plastic plate adjustable, meaning make a plate that does not wrap around the four screws.
See green boundary in picture below showing the outline of the plate.
Second, if you use the plastic tape/nylon spacer top stabilization method, the plate will not fit through the spacer. The plate should fit if you use the screw insulator top stabilization method, but even with the screw insulator, when you put the bottom stabilization plate into the shaft, the center hole will have to go through the threads firsts, which actually widens the center hole so that there is still some, albeit slight, horizontal movement when you begin grinding. I think the shaft underneath the threading is actually only 7.5 mm, whereas the threading is 8 mm.
This is my modification that seemed to work better for me. Metal plate is not an option as it seems to involve too much work and I have no drilling equipment. I used plastic like AD's method, but the shape of my plastic plate resembles EricBNC's method as I like to have more of the plate in contact with the inner ring once I screw everything back in. I think a wider plate like EricBNC's provides more stability to the hole.
First, sketch out how you want the plate to look like on a piece of paper (actual scale). I first used the disassembled metal ring where the four screws go in, and trace both the outer and inner circle. Second, while the metal ring is still on the paper, I put a "dot" in the screwing holes indicating where the screws are located. Now take out the metal ring, and what you have on the paper is your "working area" indicating the boundary limits to the size of the plastic plate.
Still with the paper, connect the "dots" diagonally, and the center dot that is created shows where the center hole should be located on your final plate. Now draw out the rest of the plate on paper so it resembles the green boundary in the picture below.

Next, you need to transfer your "plate sketch" to the plastic you intend to use for the plastic plate. Before doing that, you need to make the center hole in the plastic first. I found a semi-hard flat plastic cover with enough flat area from my food container and created the hole by melting it with a hot 7.5 mm drill piece that I heated over the stove wearing gloves to prevent burns. The plastic should not be too pliable as it would not be stable and firm enough to prevent the shaft from shifting horizontally when grinding. But it can't be too hard or else shaping the hole would be difficult with my melting method. Once you get the center hole, take care not to trim/widen it too much at this point as you can always do that later.
Once you have the center hole on the plastic, put it on top of the sketched paper and center the hole accordingly with the center dot on paper. Hopefully, you can see through the plastic and into your sketch. Now trace on the plastic, and then when done, cut it using scissors.
Now that you have a plastic plate cut out to size, cut a slit so you can slip the plate into steel shaft without going through the threading. Should fit tightly. If too tight, widen hole slightly.
See picture above. At first, I was worried that the slit would affect the plate's stability or fit of the centering hole, but in fact, it does not.
Now just reassemble everything by putting the steel shaft (modified with the plastic tape spacer and with plastic plate slipped on) through the outer ceramic burr, and then through the tower. Now position the plastic plate accordingly making sure there is wiggle room to reposition the center hole if needbe. Then CAREFULLY put the metal ring over the outer burr, making sure the ALIGNMENT is correct and put in the four screws and then tighten the screw.
(I say this because if the alignment is incorrect, you will slightly crack the outer ceramic burr when tightening, which is what happened to me. I used Super Glue to glue the cracked piece back into place.) Now put the handle and everything back together on the top, and test it out.
If you're like me, you will find that with this mod, it is super stable and there is virtually no more horizontal movement flaw that originally came with the Skerton. However, there is a new problem: the burr grinds toward one side. As I mentioned at the beginning, there are several reasons for this and none of them are definitive. To fix it, slightly loosen the four screws, and reposition the outer ceramic burr accordingly as the outer burr also can also move around slightly within the metal ring. And then tighten the screws. If the problem persists, then use a screw driver or something, and reposition the plastic plate accordingly and tighten the screws.
Centering is obviously the hardest part, because there are several things that work against centering even once you stabilized the steel shaft. This is why my mod makes disassembling and cleaning the Skerton quite a chore; once you disassemble it, you will need to spend time re-centering it all over again. The first time, it took me about 30 minutes to find the sweet spot, but it was worth it as the grinds are super uniform now. Will take picture later for y'all with my friend's camera. I intend to clean my Skerton only once a month, so I don't mind the trouble.