If you take a look at that sheet, there really isn't a lot of variance apart from clipping diodes, inversion of the TONE/FILTER wiring, those necessitated by choice of IC, bypass switching systems, coupling caps without appreciable changes in frequency response, and a pull-down resistor at the input. (Certainly not the wide variance seen with the BMP.)
In terms of mods, the Ruetz mod ("sweep" on the Aion board) or
@Chuck D. Bones'
CONTOUR control are useful and can be taken out of the circuit entirely. Clipping options are an obvious choice: I prefer 2x1 Si diodes (e.g., 1n4148 or 1n914), but usually include 1x1 LEDs (Turbo) and 1x1 Si (stock) as well with an On-Off-On toggle.
The 'White Face' small box Rat gets a lot of attention on the internet. It's the V2 FILTER control circuit. That being said, most Rat versions are the same circuit in different housings or aesthetics and claims about sonic differences are specious. The real change comes with the introduction of the OP07 chip (see below).
The Rat is definitely my favorite distortion. Stock, it's got a ton of options and is very versatile, but it can be easily modded to adjust things as necessary. It has a usable range from OD to distortion to fuzz [1] and it stacks well with (octave) fuzzes, ODs, and boosts. Frankly, I'm surprised I don't see more boards with two Rats on them. I don't really think it's terribly productive to argue about IC choice here [2], but I would encourage you to try different chips out and see what you like. The FILTER control can be narrowed to have more precision if you find that there's only a window that works for you.
It's a simple circuit and easy to bread board. I'd suggest building one stock and modding another one out. For inspiration beyond clipping and typical tone mods, the VFE Alpha Dog is a more ambitious take on a modded Rat.
1. There are a lot of varieties of 'fuzz'. It gets fuzzy at full-bore. We don't need to have an inquisition here.
2. There are a lot of questionable devices being sold as 'LM308' chips and not a lot of control over test methodology. Some unscrupulous stateside resellers purchase bulk eBay listings and pass them off as authentic. Are they? Who knows. SBE and other sellers with good reputations can be trusted, but there is a lot of uncertainty in the market.