The first thing I would want to know is what kind of a laptop this is.
From the video or what's been mentioned, I can only gather that it's an acer. But what kind exactly? (e.g. "travelmate P543". This should be on a sticker on the bottom of the thing, or right click 'my computer', then 'properties' and check 'System'...it should be next to 'Model').
With this information, you can probably go to
acer support page and download the latest BIOS for the system. In turn, you can use a spare USB key to make a flashable USB drive.
I admit: the hard part will be to get that USB drive to actually flash the BIOS. If you press F2 during boot...you say it shows nothing. But at that time, does it continue booting?
If not, that's actually a good thing. It can be that it's "just" the video card drivers that aren't recognized during BIOS(1). In that case, the menu options are still there. It's just a matter of figuring out how many times to press the arrow keys before pressing the enter key.
EDIT: upon rereading, I see you haven't really replied to the "remove the battery" suggestion. This should be something like this on the motherboard:
(I obviously don't know all brands, but thus far I've seen this battery on most laptops I've opened up(2) )
(1): I know I'm repeating some other mentions, but...you've tried booting with a secondary monitor connected, right? (preferably VGA, but HDMI if your model doesn't have that connection).
A different situation as an example: one of our clients managed to give himself driver issues to the point where he couldn't see anything. It could, however, still do everything else. I just took over the PC through teamviewer and had no problem seeing his screen that way (no, that won't work for BIOS related issues)
(2): okay, okay...I've seen a few HP laptops that simply had NO way to properly open up things. To my knowledge, acer has never done that sort of shenanigans.