I realise that with this antenna design part of the coax is used as a counterpoise, but does it matter at 10mw?
The power doesn't matters. The antenna geometry matters.
Common mode choke affects your antenna geometry, because it prevents to use feeder, equipment chassis, mains line and other things as a part of antenna.
So, it doesn't matter if it's TX with 1000000 W, 0.001 W or 0.0000000001 W, or if it's RX signal.
In any case you're needs RF choke to keep proper antenna geometry.