I have seen other arguments over this. If the caption contains ONLY Tayler and ___, then I would use "me" thinking it was a shorthand way of writing "This is a picture of Tayler and me."
Of course, if you add to the caption ("Tayler and I take in the sites" or "Tayler and I enjoy a chocolate croissant" for example) then your "remove Tayler" rules works to remind you to use I in situations like that.
[a writer, not a teacher]
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