Thanks for the tip. Ordered a female TRRS jack and a male TRRS plug as well as a 2-3 ft. TRRS to TRRS cable from a different source (ridiculously expensive shipping, but was pretty much the only choice).
Man, what a difficult solder!
Cutting the PC360 cable and trying to get those tiny thready wires into the female TRRS jack was rough. It's hard because each conductor is just a few strands thin of copper wire surrounded by a nonconductive varnish (instead of insulation to prevent shorting between the wires), making them very resistant to accepting solder. Also each conductor bundle seems to have a core of a few thin fibers (kevlar?), making it hard to cut cleanly, and the fibers interfere with getting a good solder too. For a little while I almost thought I had junked the headphones, and figured it was a lost gamble and I'd need to buy a new set. I even tried to get inside the earcups to see if I could just solder new cable right from the source, but couldn't find a way to get in those.
But, I soldiered on and finally got it soldered up and when I plugged it in... HOLY CRAP it worked! Seriously surprised (and relieved!) that it worked! In fact it works really well, the only side effect is that when I turn the volume all the way down on the mixamp, if sound is playing I can still hear it albeit very faint. I think it must be some kind of crosstalk or bleed due to the mangling of the conductors from the soldering. But I don't think I can make it any better than it is, and there is not enough spare cable on the headset side to attempt a fix only to have me screw up and junk the headset.
The male TRRS plug I bought I was going to turn the rest of the cable that I cut from the PC360 into a dual TRS to TRRS male-male cable (basically exactly how it comes stock) so I could use the PC360 on the PC. That solder job was even harder since the leads on the male TRRS plug were ridiculously difficult to attach (Again with the varnish making the little conductors not easily accepting solder), and after several tries I finally thought I had it, but when I tested it on the PC the mic didn't work and also the sound balance was off (left channel was okay, but right channel would play out of both channels simultaneously). I am debating whether to order another style of TRRS male plug to try the soldering again, or just buying premade cable to do the job. It's kind of embarrassing for me to say since I usually consider myself competent at soldering, but this is almost beyond my ability (or perhaps my equipment, I don't have any of those little holder clips and stuff to keep things in position) to confidently do a good job at. The plug I got was a "low profile TRRS" and I think if I got a regular one, it might have more traditional leads that would be easier to solder onto like the female TRRS jack I bought had.
FWIW, if anyone else has the guts to tackle a cable mod on their PC360s, if you snip the cable you will see a total of 5 conductors. Red, Green, Red+Green, bare copper (strands that surround a white insulated conductor), and a white insulated conductor (which has another bare copper strand within). The pinout is as follows:
Green- Audio L(?) <-- goes to the tip of the headphone jack
Red - Audio R (?) <-- goes to the first ring of the headphone jack
Green+Red - Ground <-- goes to the sleeve of the headphone jack
Bare copper - Ground for mic <-- goes to sleeve of mic jack
Bare copper inside white insulation - Mic signal <-- goes to first ring of mic jack
To make the connections to the 4-conductor jack I joined the two grounds together and used the following:
tip: green cable
first ring: red cable
second ring: green+red cable, plus bare copper ground
sleeve: mic conductor (white insulated cable)
Anyway tl:dr- I got the headset soldered correctly, and now I am happily enjoying having my mixamp rx on a short leash, no more miles and miles of spare cable washing around! Need to address the PC connection so I can use the headset on the PC still though.