Well, I would find it really awkward for a Bayesian to condone a modus operandi such as "The p-value of 0.15 indicates it is much more likely that there is a correlation than that the result is due to chance, however for all intents and purposes the scientific community will treat the correlation as non-existent, since we're not sufficiently certain of it (even though it likely exists)".

The way statistically significant scientific studies are currently used is not like this. The meaning conveyed and the practical effect of official people declaring statistically significant findings is not a simple declaration of the Bayesian evidence implied by the particular statistical test returning less than 0.05. Because of this, I have no qualms with saying that I would prefer lower values than p<0.05 to be used in the place where that standard is currently used. No rejection of Bayesian epistemology is implied.