You are probably fed up with suggestions by now, "but", if you have jumpers on your motherboard this may work.
The third paragraph could be of interest to you.
Hope it in some way helps, good luck and post your results.
Using the jumpers
The canonical way to flash the BIOS via hardware is to plug, unplug, or switch a jumper on the motherboard (for "switching a jumper" I mean that you find a jumper that joins the central pin and a side pin of a group of three pins, you should then unplug the jumper and then plug it to the central pin and to the pin on the opposite side, so if the jumper is normally on position 1-2, you have to put it on position 2-3, or vice versa).
This jumper is not always located near to the BIOS, but could be anywhere on the motherboard.
To find the correct jumper you should read the motherboard's manual.
Once you've located the correct jumper, switch it (or plug or unplug it, depending from what the manual says) while the computer is turned OFF.
Wait a couple of seconds then put the jumper back to its original position. In some motherboards it may happen that the computer will automatically turn itself on, after flashing the BIOS.
In this case, turn it off, and put the jumper back to its original position, then turn it on again.
Other motherboards require you turn the computer on for a few seconds to flash the BIOS.
If you don't have the motherboard's manual, you'll have to "brute force" it... trying out all the jumpers.
In this case, try first the isolated ones (not in a group), the ones near to the BIOS, and the ones you can switch (as I explained before).
If all them fail, try all the others. However, you must modify the status of only one jumper per attempt, otherwise you could damage the motherboard (since you don't know what the jumper you modified is actually meant for).
If the password request screen still appear, try another one.
If after flashing the BIOS, the computer won't boot when you turn it on, turn it off, and wait some seconds before to retry.