Without reading ... nuclear materials have a half life and beta decay is a nuclear process. The battery will decline in a known curve.

Beta emitted electrons are high energy and there may be some erosion of the zone between the two materials which would reduce efficiency. Your mileage may vary.

And one either has to mine beat emitters or make them using a nuclear reactor ... oh, just followed a link - use tritium for this? Sure ... we're talking long term spaceship batteries that will be less controversial than boosting bits of plutonium into orbit, but that is the only use such a technology will see as far as I can tell ...

Electrons from beta decay of tritium have an energy of 5.7 keV with a decay constant of 1.8E-9/sec. If one had a kilogram of water with both hydrogens as tritium, this leads to (starting out) a power output of just over 100 watts, which decreases to 50 watts after 12 years. Useful perhaps for the Eveready Energizer Bunny. Since it runs non-stop whether you use it or not, one could trickle-charge a conventional battery and make it power your toaster occasionally. All this assumes, however, 100% conversion of the electron kinetic energy into useful power, which is of course ridiculous. And as others have pointed out, only around 250 kg of tritium (equal to about 1500 kg tritiated water in my example) have been produced in the US since 1955.