I found out from a fellow Techlahoman on the slack that those connectors in my last post are JST-SSH. Luckily, Digikey carries both genders of these connectors. They also carry some SSH jumpers that can be hacked up for our purposes. I ended up soldering a 6 pin header onto the board to match the other JST connectors.
I also prepped a few sets of cables with JST "connector housings" on one end and male header pins on the other end.
In other news, in doing some digging, it appears as though the Cypress CY7C68013 on this board is connected A0-A15 to the flash chip, which has 8 times the address space of the 8051 in the 68013. To that end, they have some GPIO stuck to the highest three bits of the flash.
The EA pin of the 68013 is also tied strongly to ground, which means it's using the internal 16KB of RAM is being used for code and data. I also found that the I2C bus was connected to a 64KB SOIC flash chip. I hot-aired that off the board and wired it up to some ribbon cable with a 10pin IDC connector on it that fits directly into my BusPirate 3.6.
Dumping the chip seems to give me 4 copies of the same 16KB of data. I'm not sure if they've programmed it 4 times or if I have the part number wrong and it's only a 16KB flash chip. In either case, I'll be trying to determine what the code is doing.