MacBook Pro shuts off on battery power with new battery

I recently replaced the battery on my Early 2009 MacBook Pro as I was having issues with it shutting off without warning when running on battery power. However, this hasn't resolved the issue and it continues to only work when plugged into AC power.

I've reset the SMC multiple times, PRAM and all the other basic things as far as I know, so suspect it may be a short on the logic board or something along those lines. I also tried unplugging all connectors from the main board other than those absolutely necessary and seeing if it still shut off on battery, which it did, so don't think it's anything to do with ports, keyboard, optical drive, etc.

What ways can this be fixed? Will it have to be a straight replacement of the logic board, or is it possible to track down a short and resolve it?

You’d really need to have current / voltage probes to tell where the fault lies. In a shop with resources to order test parts, you’d still try a second battery before doing a logic board swap if you don’t have the tools or skills or documentation to diagnose the precise voltage levels in and out of the power transformer(s) that are driven by the battery.