I fully understand....

try the other battery first...

in fact, try hooking the battery plus to the end of the ballast where the brown wire is connected...btw, does the black wire that connects there go to the coil positive?

And ground the battery negative to something convenient under the hood...engine, sheet metal, regular battery clamp, etc.

If practical, disconnect the brown cable from the ballast and replace it with the jumper from the new battery...just be sure it still gets power to the coil via the black wire if that is where it goes...in other words put the power from the new battery thru the ballast and to the coil