One of the batteries (the PRAM battery) is of no consequence for power on, but the second battery kickstarts the PSU and must be working. However there are easier workarounds for testing than soldering a new one in. I ran my mac II with two 1.5V AA batteries in series for a while in place of this battery without any problem (and a completely dead PRAM battery) so 3V is sufficient. You could try the same setup. Just connect ground to the chassis and momentarily apply the 3V to pin 15 of the power connector. Once started, voltage to this pin is maintained by the PSU itself. To shutdown, simply ground this pin. (You can find pin 15 since it is the last one on the side with the missing pin (pin 13).

However, if you are getting 10 seconds of power, the battery is doing a good enough job to get the power supply started and your problem is something trickier than simply getting the PSU started.

An old tech note talks about a thermal sensor that shuts down a Mac II if the temperature gets too high. Perhaps there is a fault in that circuit? Or maybe the PSU is dying and just cant handle any load?