Of all my works in progress, I decided to finish and release this. And then I forgot to post it here.
This mod turns F12 into a key to take a screenshot without the HUD. It will disable the HUD, take the screenshot, enable the HUD again and save the Oblivion.ini, all in a few frames. It's because I'm too lazy to type "tm" into the console, yea. Saving the ini is done so you don't lose the screenshot index if the game crashes.
This really isn't much of a mod, the OBSE check script is probably longer. I am just lazy and perhaps someone else is, too.
You can change the key by entering into the console: Set HudlessScreenshotKey to 88
Where you can replace 88 with any key from this nice list.
The script runs "only" once every 0.1 seconds, so a super fast tap on the F12 key might not trigger it.
2020-06-03: I just now got Windows 10 (I know, I know, I am late to the party) and I noticed the game would hang for seconds after taking a screenshot. I had to add the Oblivion ini as an exception to Windows defender, to stop that overzealous little shit from monitoring it every time it was saved after taking a screenshot.