I think you'll find that most folks using a pull-pull or spring-pull setup are using low stretch fishing line for the pull lines such as Spyder Wire... which isn't wire but a unique plastic fiber. But if you do insist on something of the sort then I'd look into their smaller sizes. The loads on the surfaces are light by any standards. Their smallest .010 size would be more than strong enough and easier to work with for forming small neat tidy ends. And with using the smaller size you can use small lengths of 1/16 K&S aluminium tubing as the crimp sleeves very easily for very tidy looking ends.

For guide tubing look at the various small sizes of teflon tubing from the usual online shopping sources or electronics parts suppliers. If you go with the .010" size wire or with the Spectra or Spyder Wire synthetic lines either will fit through 22Ga ID tubing.