The answer to my question about which template is being rendered (at least in my case) was page.php.
That’s page.php from your theme.
So if you, like me, want to do things to the main template in which the bbpress forums are housed, you can use page.php.
If you, like me, want to do things to that template ONLY for bbpress stuff, but not for other “pages” in your site, you can use the php function is_bbpress() in that template to render conditional html based on whether it is a bbpress page or not.
This seems like amazingly basic and obvious information that I was only able to discern by gritting my teeth and putting bbpress into the text editor and reading through A LOT of code. If this information is documented somewhere easy to find, please enlighten me so that i can perform the requisite face palm. If it’s not, why not?
This is untested by me, but digging through some more code, it looks like you might be able to create a custom home-base template for your bbpress forum by copying the original page.php into a new file and naming it any of:”
'plugin-bbpress.php',
'bbpress.php',
'forums.php',
'forum.php',
'generic.php'