You might try a "fish tape." They are used by electricians to tread a
line in conduit. There are two types I know of - steel and nylon. I have both, but haven't used them enough to say which would be best in your application. The nylon fish of course is non-conductive, costs more I think, and seems a bit more flexible than the steel fish.
I've never done this, but I think you'll need to thread the fish up the mast, and probably climb the mast to retieve the end, or climb the mast and thread the fish down. Then pull through a light
line that in turn will be used to pull through the halyard. While you are at it, pull through two light
lines, and leave one for the next time it's needed. If you start and end at the mast step, you can tie off the spare
line there. My rigger did that for me during commisioning.
Fish tapes come in various lengths, and both were available at Home Depot when I bought the nylon fish.
As for climbing the mast, again I can make no comparisons, but I've climbed the mast twice to retrieve a halyard, and the ATN Top Climber worked quite well and seemed reasonably safe used as directed. ATN on the web has a video showing how it is used. Beware that the demonstrator must have practiced tens of times before the filming. It worked essentially as advertised for me but at probably about one-tenth the speed and ten-times the apparent effort.