Question: Are science and spirituality mutually exclusive?

They necessarily utilize different toolsets.

Science counts on repeatable, measurable results using (primarily) hardware independent of human experience. Spirituality references data points that are rarely repeatable, and counts on the experiential/sensory understandings of the subject. While much can understandably be said about doubting the spiritual for this reason, there ARE, within certain “experiences” (the non-dual experience, in particular), examples of repeatable data that can be evaluated by those with appropriate understanding. Science would largely dismiss such evaluation, and, in fact, such understandings are not really readily verifiable by science (though increasingly many of the conclusions neatly fit many scientific theories arrived at entirely independently).

Science, on the other hand, commonly uses the abstractions of measuring devices and, at the most basic level, admits that the evaluator is always a factor in every experiment. There is also a common hubris that would have one misapply the tool of science where is isn’t necessarily germane or suitable.

(See: Scientism - Wikipedia)

In my opinion, this misapplication of science to deprecate experience is neither useful or helpful.

The appropriate tool to evaluate mind is... the mind. Buddhism is comprised of a number of thought experiments and value propositions that can be explored using the mind, and answered, with diligent practice. This is the path I would recommend for those wishing to investigate the nature of things.