I don't see why that would happen, since the universe has already existed for billions of years. Wouldn't the transition either have happened long ago, or be so smooth that the probabilities are essentially constant within human timeframes?

Yes, realistically. You'd have to have long term horizons, or odd circumstance, to get that kind of behaviour in practice.

I don't think the law of physics postulated above would provide any evidence that you can bet on.

I'm not sure - see some of the suggestions by others in this thread. In any case, we can trivially imagine a situation where there is relevant evidence to be gathered, either of the observational or logical kind.