The SPKR jack is designed for powered speakers. If it produces a clean sound then I'd guess the TX amplifiers are somehow picking up the 1kHz from the USB cable. I've never seen this happen so don't have a specific solution but there are some things you can try.
Put a ferrite choke on the USB cable or find a USB cable that has one. Check for ground loops. Make sure you power supply is floating and doesn't have its negative tied to ground. Cut the USB_SHIELD jumper and replace it with any of the leaded capacitors you have left over from the kit. Rebuild the three transformers in the transmitter. And, as Rainer said, check the power supply voltage while transmitting to make sure the voltage doesn't drop.