I’d really like to help you but either you are really confused, or I’m really confused.
Swap out the ring for the xEM reader antenna and/or install the antenna into the wing mirror. or the @TomHarkness antenna when that is available.
You can’t just swap antenna between systems. Antennas are designed and then circuitry is selected and tuned for that specific antenna design. If you change the antenna, you also have to redesign the associated circuit.
There is a voltage difference from ~2.3V (Idle) - ~4.4V ( fob within approx. 100mm )
Where are you reading this? You can’t measure the voltage on the antenna itself. If you are, that’s the 125khz 'carrier" you are reading and will be there any time any tag is presented.
I then dropped the voltage it would cut out at 3.5 Volts,…
I’m failing to see the point of your low dropout test. The relay is meant to be powered by a constant power source, not a changing one.
You don’t want a 3.3v relay. You want a 12v relay.
The trigger on this particular module is on the ground ⏚ , I am pretty sure the antenna coil is +ve, so I need to confirm this.
Again, you can’t use the actual antenna wire to control anything.