I've updated my article along with a credit to you. Unfortunately, I couldn't figure out the other constants despite a whole lot of brute-force checking.
There used to be a site called the Inverse Symbolic Calculator (also a sibling site, Plouffe's Inverter) where you could give in a decimal expansion and it would search for known constants. Unfortunately, it seems to be down (the frontend exists, the backend just gives 404). Maybe someone else made something similar?
The ries program is very cool for turning a number into an equation. However, I tried it on some of the Pentium constants and it didn't do very well. In particular, it seems to be allergic to numeric constants, preferring complicated symbolic expressions. For instance, if I put in 0.5625, I expect to get 9/16 out. However, ries doesn't come up with that, instead suggesting the best answer is sin(pi*sqrt(x)) = 1/sqrt(2). It's impressive that it came up with that formula, but it's not helpful.
Yeah, RIES won't work well with that because there are too many constants to test out for RIES. AFAIK RIES doesn't natively handle constants larger than 9 partly to reduce the already enormous search space.
Edit: I found https://mrob.com/pub/ries/, which may or may not help.