I focused on initial braking distance because that was the scenario you presented. If you have to do repeated emergency braking on the street, then your problem is not your braking system.
On track you can get surprisingly far with OEM brakes as long as you upgrade the brake fluid and use properly bedded race pads. I really recommend you try that first. On the other hand, if you use street or sport pads with a BBK kit, brake performance will still decrease after only a few laps on track.
If your pedal feels mushy, it is because of your brake fluid. You will experience the same problem with a BBK if you use the wrong brake fluid, or if you do not bleed/change it often enough.