I'm in favor of compromise, but why the 12th inning? Haven't looked at the data but from memory it does kind of feel like games that get to the 12th inning tend to last into, say, the 15th or later.
I think 12th based on (and I cannot emphasize this enough) vibes instead of data would be fine.
10th definitely feels too soon (it's basically the 9th), and the 11th still kinda feels too soon too.
If anything, I'd argue it should be fine to ask your closer/reliever to pitch an extra inning (the 10th) "as-is". The 11th makes you burn an extra reliever, and that should be okay.
The 12th is where I'd start to say "okay, wind it down, we're all losing now".
