i make my own lanyards from quick and easy 3strand, tails too, so any problem with them is easily fixed!
i'm with letting it wear of the line too, but will try not to use new rope. Also, will try to be very concious of not letting sap get on rope.
Especially in hot florida sun, it does run out quicker as it thins! The moss down here seems to soak it up some, so i try to use that as a buffer, but keep it out of your friction hitches. Use pulleys on the upper anchors so that lowering line doesn't cut a line into bark, draining sap onto rope, and in a pinch, put a wad of moss on the anchor/frictional redirect.
We also try to complete the work on each tree; ie. don't go to lunch on a hot day, and leave part of a pine standing (or logs laying), bleeding sap out to get all over you and everything when you return.
Dawn dish soap is pretty good on getting light grades off hands etc., have tried some on ropes, but generally go with limiting exposure and wearing off, sometimes replacing tails and lanyards.
For hands we have a real sweet local citrus product down here; called 'The Scrubber' it is more expensive than other mechanix soaps, and for my money very superiour! It melts pine sap off hands like nothing i have ever seen; especially if you follow the directions.