"To ground" or "not to ground" a coax to prevent RF ingress?
The problem is of common mode currents in the coaxial feed line, induced by strong local AM radio stations.
It could be addressed by using a balanced system with two coaxial cables serving as a twin line feeder. The braids would need to be interconnected at both the ends and earthed only at the receiver end.
The receiver and sensor would require balanced input and output matched to the feeder impedance that would be double the characteristic impedance of the coaxial cable used.

Attempts by others to resolve a similar problem, through the use of a common mode choke, isolation/matching transformers and selective grounding, have yielded favourable results.
The details are accessible at
https://groups.google.com/forum/#!topic/sci.electronics.design/hKwMXrD9ksE and http://theradioboard.com/rb/viewtopic.php?f=10&amp;t=9357&amp;p=92911#p92880
It would be worth your while to explore the possibility of application of similar techniques to find a solution.
Should that be successful, the major changes required to convert your system to a balanced one, using a dual-coaxial cable feeder, could be avoided.