Is there a way to remove Battery connector from the logic board?

If you want to check the fuse, you'll need a multimeter on continuity mode. The fuse is labeled... TH or something on it. It's to the left of the white plugged spot for the charger (follow the yellow charger to the area its plugged into). You can then check the fuse there. This is a horrible image but this is the general area you want to look for and test the fuse: Fuse Location (The image is of a 1000 model but the idea is the same for location purposes).

The connector can be desoldered and another one attached. If you're good with a soldering iron you could attach another one in it's place. Chance are though that it's a fuse issue. Check with the fuse first before you go off destroying the charger pins to the PCB.