You can't "bypass" the CPS. It would be like trying to bypass the points inside a distributor. Also, the CPS has nothing to do with the starter circuit. You could remove the CPA and the starter should still crank the engine.
As to the original damage: Where did the battery cable burn off? To burn through would require a dead short, high amperage draw, or a bad (poor contact) terminal. Did it burn one of the ends off, or somewhere in the middle?
From what I read, you can't get it to turn over, let alone start. Will the engine even turn? Put a socket/breaker bar on the crank snout bolt and try to turn it. The crank should turn. If you can't turn it by hand, the motor's locked up.
Electrically, start with the obvious. Do you have a multi-tester? If not get a cheap one. Hardware store item - 15 bucks.
1) Does the battery have a charge? Check it. A fully charged battery will read 12.2-12.6 volts.
2) Is there power at the starter relay?
3) Does the battery have a good ground? Check both to the engine block and to the chassis.
When you say "no crank or anything at all", do you get dash lights when you turn the key? When you try to start in (with the key, or jumping the starter) do you get nothing, or a loud "click/hum" with no movement?