Now this is interesting. You replaced the BIOS battery and it restarted and functioned 1 time? When you remove this battery it resets the BIOS to default conditions. When it starts up the 1st time, it looks for RAM, drives, etc and stores this info in the BIOS CMOS. No, this battery is not like a fuse.
It's possible that some bad component is preventing the computer from starting. Remove any add-in cards and the data & power cables from the hard drive, CD-ROM, etc, so the BIOS won't see them. Remove all RAM except for 1, some mobos won't do anything without some RAM. Then see if you can just get to the BIOS (setup) screen more than once. This will prove it's not the motherboard. Then, by deduction, the culprit is one of the devices you removed. Start plugging them back in, one at a time, until it fails, and you've found the bad one.
BTW, you can check the BIOS (CMOS) battery with a voltmeter. It's probably a CR-2032 and is 3.0VDC. I suspect the old one is just fine.