^^OK, there are maybe forty different isotopes of Radium and a very few go through "Isomeric Decay" that may emit a high-energy photon (gamma emission) but these isotopes have half lives in the milliseconds. You ain't gonna find many of them.
Alpha particles can't get through the layer of dead skin we all wear. I really doubt it can get through the case of that detector.
OP, why don't you PM it to me. I have some old watches and aircraft instruments that are guaranteed to contain radium. Can't say I need one bad enough to spend a couple hundred on it tho...
The reason I mentioned radium 226 is that's the isotope in the decay chain for U238, the most abundant form of Uranium. That's where most of the radium in the crust came from. Interesting, the half-life of U238 is 4.5GY. But it's had plenty of time to decay.