Here is another test you can run: connect a 6' wire to the ground side of the BNC
connector on the radio, with nothing to the center pin. Extend it as you would a
jumper cable. Do you get any interference? Try attaching the far end to the shell
of the mating jack.
If any of that causes you to hear the interference, then better shielding on your
jumper cables won't help (or at least won't make a big difference.) You've just
got too much RF around the radio.
Using a T-R switch...
What type of T-R switch? Not a relay I presume, because it couldn't operate
fast enough. So an electronic T-R switch? Does it allow the QS1R to hear
between the dits when you are transmitting? While T-R switches are designed
to protect the front end of the receiver, they generally aren't intended to
allow reception at the same time, other than to hear one's own fist in the
receiver. A fast switch acts as a modulator, so it may well superimpose your
signal on the others it is hearing because of that.
A repeater typically requires 100+dB of isolation between transmit and receive
to maintain normal RX sensitivity. It isn't trivial: rigs have to be well shielded,
DC and audio cables properly filtered, tuned cavities in the TX and RX lines, etc.
What you're trying to do requires at least that much isolation" don't under
estimate the measures you may have to take. Double-shielded patch cables
are a good place to start, but it may take a lot more than that.