Is it possible to run the fish tape up from the bottom, then attach the line, or a line you could use to pull it in the direction you need to.  Someone will still have to go up the mast, of course.

Hm.  I've messed around with the metal fish tapes quite a bit and I'd suspect that, as good as the idea is to try to run it up to the top of the mast, I suspect that it will not be stiff enough to combat gravity (its own weight) and the friction of the inside of the mast and the plastic tube that the electrical wires are run up inside. No matter what, someone is going to have to go up to the mast tip since sending a feeder line up on the end of a halyard will end up with two line on one sheave up there.  No matter what, you have to have a person at the mast tip to feed the line over the correct sheave up there.

• Someone either goes up the mast, or you pull up to a dockside building or cherry picker and tilt the mast tip over to it.
• Going up the mast is easy if you use a mast crane and snatch block down on the dock on a bollard to pull a halyard upward from the block. 
• If safety issues preclude this (my club won't let us do this for fear of insurance issues -- but if we use a second halyard as a fail-safe this seems overly cautious) I bought two climbing rope brakes ('belay devices') pretty cheaply on Amazon and use one on my bosun's chair connection with the other on a rope-step that i pull up from below the chair as I go up the halyard that is fixed to the base of the mast.  I use a spare halyard as a safety rope that someone pulls the slack up as I go up. Using the belay brakes means I do all the work and no one has to try to winch me up.
• You can then send a metal snake down the inside of the mast to the port that the topping lift goes through (use another bit of stiff wire in a loop to pull it out of the topping lift port at the base),
• Attach a sufficiently long, thin, strong feeder rope to it and pull that up to the top to attach the end of the topping lift line to it up above.
• And pull the feeder rope and topping lift line down and out through the port from below.
• NOTE that the best way I've found of securing a thin rope to the thicker line is to tie it around a medium duty zip tie, overlap the feeder about 4" over the line, pull the zip tie super tight around the line, put another zip tie on around the overlap below the first one, pull both super tight, cut off the tail close to prevent it from sticking out, over-wrap the overlap in electrical tape and away you go.

Best of luck! 

 

Someone can go up the mast and drop a messenger line with a bit of weight down. Mason's line from HD or Lowe's is bright orange, pink or green and strong enough for the job. Several 1/4-20 nuts of a small sinker that can pass over the sheave is enough weight. 
It sounds like the riggers first attempt ran down the back of the mast and got hung up on the furler mechanism. The opening for the topping lift is above that.  With bright colored string, a bit of wire to hook it and a flashlight to look in the hole (maybe remove the cover) you can probably/hopefully get it without taking down the mast.
The mast has a small tube in the front for a wire conduit and a bigger tube in the back for the furler. Once over the sheave it ought to be a straight drop down. 
If you are in a slip and have a crowd watching make them stand on the bow so the line runs down the forward side of the mast - easier to see and catch with a wire!
 

My topping lift (C36 Mk1) is just a steel cable attached to the top of the mast (does not run internally) with a short section of rope before attaching to the end of the boom.  A small pulley system mounted on the boom allows for boom lift adjustment.
 
This might be an easier solution than trying to run the line through the mast which is likely already crowded with other lines.