I figured this out, and for those wondering:
Using the image from here: mrrrk.net | Replacing the starter button on my Lotus Elise
of the connector, you will see it marked with A-D
...Where the A-D are designated as follows:
A = Bulb +
B = Bulb - / Ground
C = Immobiliser
D = Ignition
Where A is the first red wire on the Lotus plug (the rest are black, because sure, why not?).
I don't know what the deal is with A, because it doesn't appear to do anything - maybe that's just my ignorance here.
Here's the Billet Button diagram:
The top middle connector (I believe is red on their wires) can be wired to the ignition (D), along with the far right button connector (green wire - either one). This will mean the light will always be on, when the keys are in the ACC position.
The other green wire on the Billet button (far left switch connection) will be wired to the immobilizer (C).
The rest just tuck under. As for finding something to hold it in the hole? Good luck - I used a 3/4" rubber grommet on the front side, and the original plastic nut for the original start button forced on the back. (Maybe next time I should do 28mm instead of 22mm - I thought I did.)
I now have another issue - when I was stranded, thanks to the original start button not working, I had to pull some of the wires out of the side panel, break open the start button, and jump the connection using the metal on my garage door opener (The mind goes to some next level MacGyver mess, when you're stranded). Now the needles on the dash stopped working, and the immobilizer light comes on blinking while driving, and a fuse melted for the AC Compressor - replaced fuse, no issue there, but the dash is still dead. Checked remaining fuses under the dash, and the fuse box in the hood area - all looks good.