Well. It appears I lied to you (not intentionally). As i was removing the battery from the "box" that holds it and the starter button, one of the wires practically fell out of the plastic connector between the battery and the button. I was able to put the wire back into the connector and solder it secure. Now the mower starts as it should with the electric starter. Apparently the bad connection (I told you they were all good) was causing enough resistance that it couldn't start.
I'm certainly glad it was fixable as I was getting nowhere trying to find a replacement battery. A local parts house told me that, if he could get it, it would be in the form of the complete "box" containing the battery, start button, and connectors to the tune of about $200. Ouch! I was about ready to buy a 12v gel cell battery and mount it somewhere remotely from the start button. I think that would work, but I don't need to worry about it until the OEM battery actually fails.
Thanks for all the help and advice.