How can I treat and prevent rust in steel wire rope?
That horse has already left the barn.
You striped off the protective  rubberized coating and exposed the cable underneath which is not galvanized and the rust has already started to eat the cable. Had you left the coating on it would have lasted much longer, at least until UV light cracked and compromised the coating and exposed the cable to the elements.
The best cable for outdoor applications is stainless steel (especially in your location).
Also you should have use the protective Wire Rope Thimbles on the loop going around the eye bolt.
Coating the already rusted cable will be futile, the rust is already there and probably deep in the strands of wire.
I suppose you could coat the rusted cable with naval jelly and then spray a coating of liquid rubber but that is really just a hack  which is unlikely to stop the rust and you will need to replace the cable in the future anyway.