Kris, here's what I would try:

From inside the cooler, take the chuck end of the bell hanger bit (or a glowrod) and put a piece of fishing line through the hole in the bit. Push the end with the string through the hole as far as you can and then bring the bit back in just a tad creating a small loop on the line. If possible, push more of the line through the hole to make a bigger loop.

From up top, take a piece of #12 solid, strip off the end and make a hook. Use that to "fish" for your string.

Actually, I would proabaly just drill through the top of the cooler and run the wire that way. Would that be possible?

-Larry