Do you have concerns that the sheave might lodge in a position that will catch/abrade intenal halyards or electrical wires?
One thought: Buy five 10' lengths of say 3/4" PVC pipe. And the joining coupling fittings. Glue up to make a 50' ramming rod? Maybe by feel you can sense when the end comes up to the sheave and push it out? Before inserting, maybe bevel the 90 degree edges of the pvc joining fittings to a more subtle angle to avoid the couplings from getting irrevocably stuck on something inside the mast such a protruding bolt (say) like around the spreader.
If not already in the position with forward end of the mast facing down towards the ground, then maybe rotate. The sheave might then stay in the downward curved "trough" as it is pushed.
Whoops forgot to ask if the mast head sheave fitting piece is removed? If not, probably no opening up their wide enough to enable the PVC rod to go in.
Think this through first!