I'm not saying there's a 95% chance the Higgs isn't there. I'm saying there was some non-zero probability that it was in the specified energy range before, and given these results the probability that it's now in that range is now 5% of the value it was before. Or more precisely, the probability that we're in the universe where the Higgs is in the specified energy range and it's detectable with this experiment just went down to 5% of it's original value. Even if we can't accurately identify what the value of the probability is, we can still say it went down to 5% of its original value on the basis of the evidence from this experiment. I would also argue that prior to this experiment the probability of the Higgs appearing in this energy range given our state of evidence of how the universe works was significant (more than 15% I'd say) otherwise we wouldn't have built the LHC. I understand you can't say this with a frequentist approach because we don't have a sample of universes to draw on to estimate priors.

Finally we're not talking about the probability of some state of the universe in the absolute (as frequentists claim, we can't meaningfully talk about such things), we're talking about some state of the universe given the evidence we have.