You're not discounting the idea that humans have experiences for which they feel they can only describe using culturally-derived religious or spiritual language.
Well that reminds me of Lucretius talking about the poor depth of Latin in comparison with Greek! I am not sure I would agree with that, though, I think we re talking about experiences which can or should be able to be put into words. You way be correct that people who are interested in religious and spiritual matters have refined the terminology in ways that isn't common, but I would still expect to be able to communicate in normal words about the experience.
As for this:
Can Epicurus's teachings provide an alternate framework within which to interpret these real experiences without denigrating or belittling the person who experiences them?
Yes, that's what I think we are talking about. We have some pretty interesting Epicurean texts on the subject, but they are not clear on the details of the experiences they are talking about, and for all we know the experiences that today fit under the term "spiritual experience" might or might not be totally foreign to what they were referring to.
I think my best comment is to continue to say that we're flying blind unless we discuss particulars.
Here are a couple of other related thoughts in the form of questions:
1) In this discussion are we suggesting that there are characteristics or hypothetical interactions with us which are in any way excluded from "scientific" examination? Is anyone suggesting that this area is prima facie off limits to "science?" If so, how can we even engage in conversation about them, so I presume the answer to this is no?
2) If we agree that what we are talking about can be systematically studied, would there be a way to eliminate the possibility that the experiences we are talking about are coming from within the brain rather than from outside?
3) I personally hold open the possibility that there are all sorts of "natural" phenomena that are not yet recognized, just like radio and X-rays were at one time not recognized, and (to my understanding "gravity waves" are accepted to exist but are still not understood.) However if we accept for the sake of argument that such a phenomena might be involved here, should we not presume that such phenomena will at some point be just as capable of being studied, an analyzed as accurate or distorted, as the other phenomena we are currently familiar with? No one is suggesting that there is completed information/opinion being deposited directly in the brain in fully-formed completion, correct?