Hi GaryS. I have the same 2003 K1200GT bike. I replaced all the brake lines with steel for my own handlebar riser extension project.
That said, I placed air in all the lines everywhere. It was a challenge to get the air out but eventually, I got solid brakes. Keep reading up on the ABS bleeding process. I succeeded by doing the typical normal bleed as if all the lines were already full and just pushed the fluid out the bleeders.
youtube on the k1200RS is the same bike.
is a good video part 1 and part 2
Started with front brake bleeding, full reservoirs in the back inside container and also the handlebar. I had the special $40 funnel with the long neck but you can do with a baby eyedropper to top it off. Front bleeding uses the servo motor.
Rear brake being dry was a problem but when touching the rear pedal, a servo also pumps fluid out the rear caliper along with the master cylinder action on the pedal. They activate together. The rear pedal was limp and wouldn't firm up as expected. That stumped me to the point of buying a replacement master cylinder that I probably didn't need.
Anyway, I moved on to the ABS, starting with the Fill connector. You need a quick bleeder on there. Also, the battery is out to get access and prevents the servo pump from running. You want this a manual process.
Front bleeds are this order with a repeat. F1, F2, F3, F1 bleeds. Had to bend a Menards 7mm box end wrench to get in there. A socket to loosen it and the box end bent 90 degrees at the head to open/close the bleeder while hand pump the front brake to push fluid through. Caution, it squirts brake fluid out the handlebar reservoir if you pump quickly. Gets messy.
Once front done, rear is the R1, R2, R3, R1 again. This uses the rear brake pedal with no battery servo action. Mine was near dry air in there but with slow patience, and steady "open bleeder, press slowly, close bleeder, slow return pedal", after 10 times or so, the rear pedal firmed up and I could see fluid coming out of R1. The rear reservoir tank dropped. Bubbles some. Moving on to R2, lots of bubbles. Moving on to R3, a few more, back to R1, a few bubbles. Ok to repeat the entire Rear brake pattern of R1, R2, R3, R1.
Sounds like you have some experience with repairs and patience. I had no success at using a professional brake pump/vacumm tool to reverse push the fluids through the system. Too many problems clamping hoses to the bleeders, etc.
good luck. The 2003 GT is a nice ride.