You don't need a long snake, just a new length of stiff fence wire.
DON'T use a snake or stiff wire if you have internal halyards as it is critical you don't get the new wire twisted round any halyards. With the mast down, and the halyards replaced by messenger lines, it should be possible to use an existing cable if possible or one of the halyard messenger lines to pull through a new messenger line then fish it through the correct holes with a hooked wire.
When you have a messenger going to the correct place use it to pull through the new cable AND A PERMENANT MOUSING LINE for the next time!
To stop it banging inside the mast use clusters of three small cable ties at 120 degrees to each other with the ends left long every few feet. Do NOT over tighten them as they must not distort the cable.
The oversize holes required to get the cable ties in mean you have clearance to put rubber grommets in them to protect the cable at both ends.
It will be quite easy unless you have had an idiot previous owner who has used expanding foam to stop the wires rattling . . . . !!!