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Is it possible to, somehow, access xterm's scrollback buffer as a (read-only) file or a character device?

The core issue (to avoid x/y "problemming"), is this:
sometimes the command I've just executed creates non-deterministic output, and I'd like to use its output somehow without pre-thought of tee-ing it. Right now, the only way to do this (that I'm aware of) is to use the mouse to select the text into primary selection.

You could do this by telling xterm to print the data using the print-everything action (normally not bound to a key).

Alternatively, there's an escape sequence documented in XTerm Control Sequences:

CSI ? Pm i 
 Media Copy (MC), DEC-specific. 
 Ps = 1 -> Print line containing cursor. 
 Ps = 4 -> Turn off autoprint mode. 
 Ps = 5 -> Turn on autoprint mode. 
 Ps = 1 0 -> Print composed display, ignores DECPEX. 
 Ps = 1 1 -> Print all pages.

which could be invoked as

printf '33[?11i'

But either approach (to write to a file) would need a printerCommand configured.

You can always use something like tmux which provides capture-pane that does what you want, along with splits (vertical and horizontal) and multiplexing, session support, and a slew of other things.

Or use a different terminal emulator xfce4-terminal is also lightweight and provides an easy Save Contents