Comparative Testing with a Handheld
Sorry if this has been already suggested (I don't have the time to read all the posts).
When the handheld was used was its own rubber antenna used or was a coax link used to hook up to the aux antenna output?
If the Aux Antenna socket wasn't used then could you either, do this or if you don't have a link cable (coax with BNCs professionally fitted on both ends) remove the BNC connector from the back of the radio, bring it into the cabin and connect directly to the handheld. If the noise is still absent from the handheld at all times then your Antenna installation is not causing the problem. If the noise is present as originally described then I would (initially for testing purposes) run a temporary brand new coax cable from the handheld (again with a professionally fitted BNC plug on the radio end) to the antenna connection (removing the original coax temporarily) making sure that both the centre core and the braid are not bridged by loose braid hairs using a magnifying glass and very bright torch to be 100% certain.
If as a result of this new temporary installation the radio test is clear of all interference then you existing Coax cable is causing the problem. Replace it with a brand new cable and professionally fitting a new BNC plug on the radio end.