1) I dont think there is any special issue here with discrete vs. real, because the branching ratio example could easily be formulated in terms of a simple head vs. tail example (with biased coin so that e.g. head is seen very rarely).

2) People often make mistakes about confidence intervals, when they are not careful about the idea that one tests against a (null) hypothesis.
In other words, considering an interval A < B a frequentist would (try to) rule out that the ratio is less than A with certain confidence e.g. 90% and also rule out with some confidence that the ratio is bigger than B.
If one deals with non-trivial measurement errors (as in your example) it is not immediately clear how to best 'combine' the two – and I think that is the real reason for the unexpected result of your analysis.

3) As for why (some) physicists (like me) hesitate to use Bayesian statistics, one reason (and there are several) is the danger that it blurs the line between experiment and theory.
You say it is obvious that m cannot be less than zero. The next guy is certain that your experiment *must* produce Gooks because of superstring theory and therefore wants to use a different prior from the guy who thinks that string theory is not even wrong.

One guy wants to use a uniform (uninformed) prior for m and the next guy thinks that x = exp(m) is the 'natural' variable to consider and the prior should be uniform in x.