Primary difference between the 4060 and 4061 is sensitivity. 4060 is considerably more sensitive, producing a hotter output for the same SPL.
Check the voltage the transmitter supplies to the microphone, measuring the voltage across the ring and sleeve contacts (ring = pins 2/3, sleeve = pin 4 in your diagram). It should be 5 to 10V to correctly power 4060 or 4061. 4063 has the same sensitivity as 4061 but can be powered by 3V, developed for use with wireless transmitters which do not supply 5 or more volts.
Inline TRS minijack/miniplug extension cables (headphone extension cables) work fine IME without interference, and I used those for years, but the connections can get finicky. For best performance I'd clean them with Caig and gaff tape them tightly together, but eventually they'd go intermittent or staticy and have to be redone. I've now transitioned most of my miniature DPA extensions, preamps, and PFA powering devices from 3.5mm TRS mini-plug/jack to locking mini-xlr, which eliminated those hassles.
Reterminating these mics is simple, but the center signal wire is very fine gauge and a bit challenging to strip and solder.