The CDI isn't supposed to be melted! That's your first clue that you have a problem. Numerous changes have been made to that CDI over the years from date of engine mfgr.....like my 2002. I have had several go bad on me in the 12 years I had my boat and each replacement had a higher numbered series part number....indicating a revised edition.
My test for spark is a battery powered timing light. I hook the sense clip on each cylinder at whatever rpm I chose to run and look for consistency in the readings...all 3 should be blinking with the same intensity at the same rate for a given rpm.
The melted CDI is not a "battery won't turn the motor over" root cause unless by some hook or crook, whatever it was that melted the CDI didn't pop your safety fuse but was enough current draw to discharge your battery.....otherwise no relationship between the two problems.
I would shoot a question off to Mercury Marine and ask the current part number for that engine and get one of those....even if they are pricy.....sometimes the most frugal purchase cost you the most money initially. They sell them on here. On running without that cylinder, you only have 3 and that's 33.33% of your total not counting the loading on the other two caused by having to drag around a non responsive 3rd cylinder.
