Those who say science and religion are mutually exclusive are working from the philosophical premise that there can be something outside of nature.
Those who claim that religion can be scientifically investigated, come from the philosophical premise that there is nothing outside of nature.
As neither position is superior one cannot use logic to assign greater truth.
However, the claim that there is anything beyond nature (i.e. supernatural) is the more extraordinary claim, and thus carries with it the onus to justify and explain how to reach this conclusion.
The futility of being outside of nature:
If religion is truly outside of nature it can have no measurable effect on it. If it has no measurable effect then, even if existant, it would be fair to say it couldn’t be detected by science – but then neither could it be detected by the clergy.
Thus in a non-overlapping model, the benefits of a benevolent God, such as good crops, good weather, good luck, healing or charity are impossible, as they are generally detectable.
I guess you could argue that God goes to the trouble to disguise the causes for His blessings, but why is he so afraid to show it was the result of your good faith? This argument gets a little stretched once the solutions to that are proposed. It is similar to the argument that God planted the fossils in the order of their evolutionary development to fool us into thinking that life evolved…