Further debugging:
The server running my cron job is in Europe, so today I tried it from a server in the US (New Jersey), and got exactly the same 403. I'm still able to access it from my browser, so it doesn't seem the distinction is geographical.
So the next thing I did was spoof the user agent in wget. Initially, I sent a blank user agent; that didn't work. Then I sent a random user agent example I plucked out of Mozilla doc; that didn't work. Then I actually got the user agent out of the browser that had successfully accessed the API endpoint, and spoofed that in wget; that didn't work either.
WTF.
My sweetie suggested that I actually examine the HTTP headers my browser was sending to see if there was anything there Patreon's API endpoint might be reacting positively to that I could spoof with wget. I have not yet pursued this, as I have other things I need to do tonight.
I am open to other hypotheses and suggestions.
#Patreon has not yet responded to my help request.
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