To solve the problem, I sawed open the 12k's orange nozzle assembly, as shown.

On the left of the above image, you see one half of the nozzle assembly that houses the trigger valve (detailed below) this valve prevents water from passing through the nozzle when the soaker's trigger is not being pulled.

In the image of all three parts of the nozzle assembly you also see a circular rubber piece. On an undamaged 1200, this piece is held within the retractable plastic valve, as shown below.

The fully operational valve (pictured above) should seal off the last half of the nozzle (shown below) and prevent water from leaving the soaker.

Here is an image of the rubber piece (removed from its plastic housing to allow for better viewing) as it correctly seals off the hole that leads to the gun's nozzle.

Now, here's my problem:
As soon as I sawed open the entire orange section (as seen in the first image) I looked at the nozzle and saw this:

The extreme pressure caused by the k-mod seems to have sucked the circular piece of rubber from its housing and pushed it into the tube that leads to the 12k's nozzle. This causes the trigger valve to no longer contain the pressure held within the PC and makes the gun shoot while pumping. I have attempted to put the rubber piece back into the retractable plastic section and glue the nozzle assmbly back together; this has failed twice because the same problem occurs again and again. My next plan is glue the rubber piece back into place and pray. Is this a good idea? Does anyone have any advice? Has this ever happened to anyone else?