Do not swap out the GTO-15 for coax on the run to the backstay, and do not put choke filters on the GTO-15.
The SCS Pactor modems are designed to be quite resistant to RFI, but your rig's modulator input may not be. You may want to try putting some "mix 31" chokes on that modulator input that goes from the modem
to the radio. It wouldn't hurt to put one on the coax coming from the tuner to the radio either, ideally one at the tuner and one shortly before the radio. If there is a control line from radio to tuner, put a filter choke or two on that also.
Ideally you should use a choke that you can wrap your cable through multiple times, because choke effect goes up with the square of the number of turns, or if you are using chokes just large enough for the cable to pass through, you may need several or more to do the job.
There are some good web resources for learning
about RFI and chokes. Here is one:
The Proper Split Beads to Suppress RFI
You might want to try having the ground plane attached ONLY at the tuner, and not to the radio or other parts
in the chain. This by itself might, or might not, clear up your problem.