I went and picked up a $20 digital multimeter and read the TeeBee, this is what I found:
Normal trigger: 38 mV (0.03V) with no keypress, ~12V with keypress
Inverted trigger: same as above, just, well, inverted
Resistance at "0V" (actually 38 mV): 2.4k ohms
Resistance at 12V: something way off the scale, infinite for all purposes I guess
The 38 mV seemed a little fishy to me, so I tried another gate output which measured only 0.5 mV (and goes from 0-5V) but it still didn't trigger, same resistance measurements.
I've been in contact via e-mail with Robert from Syntecno, and he mentioned there is a 1k resistor wired in series with the gate out. Since I'm seeing resistance at ~0V and the Mini wants a short, could that be what's preventing it from triggering? (I've just e-mailed Robert the same question)
While I was out, I also picked up parts to make a converter just to see if that would work (using 3.5mm female jacks instead of building it into a cable), but I wired it up incorrectly and while trying to correct it, I broke off the Base leg of the transistor so that idea will have to wait