Is it possible to change the headset cable?

Its only theoretically possible at this point! While yes the breakdown video does show the cable is removable and therefore possible to replace. The replacement parts needed have not been made available to the public. Hopefully this will change in the near future but for now your only option is to go through PlayStation support and I've seen posts on other forums saying that Sony is charging up to $300 for an out of warranty cable replacement if its due to user damage and not a defect.

I'm wondering the same thing. I've had it since launch, and my cat apparently got bored last night and started gnawing on it while I was playing. Didn't completely chew through it, but may have punctured enough for the headset to cut out. Unplugging it from the console and plugging it back in got it powered back on. Seemed to be working fine. I'm assuming it could be soldered if the damage is severe enough, but I'm not going to try that until it happens/if I know for sure how easy/difficult it is.

My dog thought I was playing with her and ran up to me and hit the cable and ripped it out. Now the USB-C is all bent to hell. I looked and it can be taken apart. I just need a new cable or maybe see if I can fix it.

It is not very easy but I have been able to repair cut cables on the psvr2. It doesn't look pretty but it works. The connector is also replaceble but it is not as strong as the original and I highly recommend using a small dongle extention afterwards to avoid straining the replaced connector again.