Les's suggestion is essentially the same as what I suggested.
SG really should be connected to chassis ground. Shielded cables are sometimes used, but rarely.
If you just do the 2/3 swap and connect 5 AND set the software to no flow control rather than hardware flow control, you LIKELY (not sure though) can send a really small job to the printer.
That's why it's so nice to have a breakout box.
Confirm again: You basically have pins on the cutter and pins on the PC so your cable is Female to Female. This generally would confirm a null modem cable.
You have no serial port documentaion of the cutter?
Does the cutter by chance use HPGL, G code or is it propreitary?
I'm trying to come up with other ideas, but there's nothing to work with.
A link to a manual even?