Make sure, that the Battery is not only 1.5V but also silver. Alkaline is not stable enough for this. AFAIK the PX625A is alkaline.
That's a good point. Your lithium cell idea looks more promising now. Not necessarily with a regulator though. I may do some experiments with an LED in series instead of the schottky diode. A 1.7v drop with a 3v cell = 1.3v so it could work.
I know the regulator is the most accurate approach but I also like to think in terms of simplicity.