The BIOS routines are not DOS - so you can't load a program to print the screen - simply can't be done.
All the software running up until a disk loads an OS is part of the BIOS software - so if your BIOS doesn't contain routines to save or print the screen (which I'm sure the vast majority, possibly all, don't) - then nothing you can do will alter that situation.
In theory DOS screens, if you run a program full screen (Alt-Enter to switch) between DOS full screen and DOS in a window - then hitting the Prt Screen key should copy the screen onto the clipboard, from whence it can be pasted into a picture or into Word or similar. If you do the same when the DOS screen is in a Window, then you'll capture the whole screen - DOS window and background as well.
I just tried this with the DOS emulator in XP and it works fine - best into Word or other text editor.
There are some programs which disable the use of Prt-Screen - but that's unusual.