First off, are you SURE it's a good battery? What you describe sounds very much like there's a grounding fault.
The inverter board wouldn't have anything to do with this unless it also failed while on only AC power, so that one's out, and same with the OS restore. The symptoms should persist regardless of power source.
If it's not the battery, my guess would be the battery connector/sleep switch. That could very well have gone bad and so when you connect the battery it somehow manages to put the system into a quasi sleep state. That part is really very cheap to get, and not even necessarily all that difficult to replace, so it would be a good place to start if you feel comfortable enough to remove the top case and take a crack at it. Just make sure you keep track of which screws go on which side of the MacBook, because the left and right sides use two different screws, and if you put one in the wrong spot, you're never going to get it out again without breaking it. One of the tricks I use, is to lay the screws out in a pattern resembling their location on the part I'm removing. And a post-it telling you which part those screws came from could be handy as well. Hopefully you never take so many of those units apart, like I do, that you pretty much have every screw and it's location memorized.
This unit starts up & runs normally with power adapter connected but if I connect a good battery,the display goes black & the sleep light blinks. I've connected an external monitor with only the power adapter & both the external & the macbook's display are fine. Again, once I connect the battery I have the external showing the desktop but there's no response when I click on any files or move a window, although I do have cursor movement. The MacBook display is black.
I've read in other discussions, that changing the inverter board, reinstalling the operating system, etc didn't work.
Please let me know what you think.