As the service manual snippet in bharvey's post shows, the Mic Audio is on pin 6 and the ground for the mic audio (MIC E) is on pin 5. The OEM Kenwood cable has a shield around the wire used for Mic Audio only. The OEM Kenwood cable lands that shield on pin 5. There are other lines in the cable that carry a clock signal from the radio to the microphone (believed to be used for the DTMF generator in the mic). Because of the proximity of the clock signal and the mic audio, that clock signal radiates into the mic audio. Kenwood went to a lot of trouble to keep that clock signal out of the mic audio, hence the special cable they use.
Ferrites on the cable may knock down RF on the cable, but do nothing to keep the clock signal out of the mic audio.
Overall shielded network cable just traps the clock signal inside the cable with the mic audio.
CAT5 vs CAT6 changes the rate of twist of the wires in the cable, but the clock signal is still there and can still radiate into the mic audio.
Changing what wire you use in the cable doesn't really address the close proximity of the clock signal to the mic audio. Telephone cables (aka twisted pair) work because twisting the two lines in the pair tends to reduce the common mode interference. Even if you put the mic audio and MIC E on two lines of a twisted pair, the clock signal is still there and the twist may not have any impact.
Your best solution is to buy the Kenwood mic extension kit. But, it's expensive and hams are cheap. There was a guy (Green Light Labs) who obtained a supply of cable just like what Kenwood uses and was building aftermarket extension cables, but he's moved on and I don't believe he is building cables anymore. His web site is still up and you can try contacting him, but you may not get a positive response.
All of the above suggested fixes may nor may not work to eliminate the ticking in your transmitted audio. If they work, great! If they don't, well at least you understand the problem.
There are two more fixes not mentioned may also solve the problem. The RJ-45 connector is susceptible to losing contact if the cable is stressed causing the connector to be pulled one way or the other. You can try re-positioning your cable to see if less stress is put on the connector. That might make the problem go away. You can also try a little bit of contact cleaner to see if that knocks the oxidation off of the connector thus improving the signal.
This is an old problem that has haunted users of the TM-V71 and TM-D710 since the beginning.