Does any man's death really diminish us as individuals?

Really? Nobody has ever been exonerated after having been convicted? Ever?

I think you might find it challenging to document that mixed feelings are a sign of mental illness. Or multitasking for that matter.

I do feel diminished by their deaths, insofar as I pay attention to them.

It’s not hard gang.

If you want to say that Donne implies that I should track every death obsessively, then I think you need to familiarize yourself with poetic license. It’s pretty straightforward to morph my interpretation into something like: “All deaths are unfortunate, even those we don’t know much about.” This isn’t exactly controversial: I support all sorts of public health measures that benefit those I don’t know or have never heard of.
ETA: Fitzgerald: “The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function.” Except we’re discussing feelings: it’s no great accomplishment to be ambivalent. But nor is it a particular concern outside of early 20th century psychology.

You defined “diminished” thusly:

I’m telling you that in the time I’ve typed this sentence, several people have died on this planet. I have not felt or perceived any loss, partly or otherwise.

Therefore, by definition, the statement that “Any man’s death diminishes us” is demonstrably untrue, unless said diminishment can be somehow offset by anti-diminishment, which you have said is not possible.

So, maybe this “diminishment” is not quantifiable, observable or even perceptible. In which case, it begs the question of how one would determine whether this diminishment exists at all. To paraphrase Carl Sagan, sure, there might be an invisible, noncorporeal, non-heat fire breathing dragon that can’t be detected in any way that lives in my garage. Or maybe there is no dragon at all.

But mankind has. Those who are gone have people remaining behind who miss them.

That may be true but it wasn’t the question. The OP asked for the validity of the statement that “Any man’s death diminishes me”. (Bolding mine).

It really isn’t that challenging.

Let’s say people are truly constantly suffering from a psychic “diminishment” every time they become aware of someone’s death. How does this “diminishment” manifest itself? If it doesn’t affect one’s well-being or their emotional state, then how can one claim it is an actual thing?

Every time I hear of someone’s death (particularly someone whose identity is known to me), I actually feel compelled to live just a little more and not take anything for granted. I don’t feel smaller. I feel bigger. I become more aware of my limited existence.

I don’t deny others might feel a subjective experience of “diminishment”. But they shouldn’t pretend they know that I experience the same thing.