It is installed!
Firstly, I want to apologize for not posting yesterday. I got the rubber on time but couldn't get make time to take a whack at it.
So a few things I learned in this process:
1. Sticky rubber that Martin (from Car Analysis) so generously provided after I reached out isn't tacky enough to pull the button. I tried multiple times by forming the rubber, letting it set for 5-10 minutes, the holding the volume assembly in (by gripping the ring, bare handed), and yanking the button with the other hand.
2. You can complete this task without removing the ring at all. This was the assumption but I can confirm.
I ended up using a tiny screwdriver. Very gently shimmied it in between the button and the ring near the L and pried. It did slip a bit but I was very cautious and gentle and the clip did let go. Then I simply worked my way around 180° (similar to how the instruction video is) while prying.
My flathead was damaged (see pics) and wouldn't be good for anything else but me being a hoarder paid off! That said, pretty sure you can achieve this with a normal flathead provided it's small enough
I took some pictures as well to help anyone who's stuck.