Even if it was possible (and it's not by following the procedure you want), I wouldn't trust such a process to be 100% reliable.
A process that is however is the paper and pen method (or even digitally photographing the screens)! Go through every screen on your BIOS settings and meticulously note down/photograph each setting. It won't take that long and you can be sure that if you upgrade the BIOS, absolutely no corruption or problem from the old settings will be imported when you attempt to reconfigure it again.
As far as fixing BSOD's try loading and saving default values and take it from there but if the problem wasn't always evident and it wasn't a change to the BIOS settings that brought it on, a change to the settings now is unlikely to get rid of the problem! If it was a BIOS settings change that brought on the issue then change it back again. You need to be thinging about what changes you did make immediately prior to the problem first cropping up.