Peter would be the guy to ask about this one, I think. I have no idea, I've never tried it. I know that modern BIOS cores have reasonably extensive USB support (at least for mouse/keyboard/disk), but I have no idea about printers. If the printer is "host-based", then there's no chance at all that would work. Many USB-connected printers are these days.
It's possible for the printer mfg to include a DOS-based USB printer driver, and a TSR program to intercept the PRINT-SCREEN keys and dump the screen to the printer, but I doubt that many would bother to write DOS programs/drivers to do that, in this day and age. Not to mention, that still wouldn't help much for printing things like the BIOS screens. So my feeling is, sorry, you're likely out of luck on this one, unless you can find an ancient parallel-port printer that works in DOS, just for the time being.