I'm having exactly the same problem as you - Vostro 3700 running Windows 10 and a prompt, by Dell SupportAssist, to install the latest BIOS update. The AC adapter was plugged in at the time and the battery fully charged. I had also selected that the PC receive power from the AC adapter instead of the battery, just in case something went wrong.
And something did indeed go wrong.
About a quarter of the way through the BIOS update, I saw a sudden flash of what I think was the Windows-stock "low battery, you might want to plug your PC in" banner across the screen and before I could think of what'd gone wrong, the screen went blank and the PC became totally unresponsive - it wouldn't even perform a hard shutdown using the power button.
I pulled the battery and the PC still wouldn't turn off. Only when I pulled the AC adapter did it die.
Now, when attempting to boot the PC with the battery in and the AC adapter connected, it powers on, cycles through the touch-responsive media keys and the HDD read/write LED flashes for a few seconds. Then, complete lifelessness - a black screen, the HDD LED remains off and there's no response to a jab of the Num Lock or Caps Lock keys, but the system responds to commands to enable and disable the keyboard backlighting by using the Fn key and right arrow key.
I've tried pulling the motherboard battery and numerous other instructions on this forum, all to no avail. Did you have any luck with any of the methods you tried?
I did more research and followed the method outlined in the link that you posted (relating to the Vostro 3500). That method must be followed to the tee, right down to the storage format of the flash drive. I even had to learn how to run an .exe with parameters to extract the .rom and .hdr files!
One more curious thing happened in my case, though: after I got into the DOS-based BIOS recovery utility and fixed the flash, the laptop booted back into Windows at the very point that it failed during the first attempted flash within Windows - it was as if it had awoken from a frozen slumber! It continued with that flash in Windows and returned an error that the update had failed, with the caution to contact my local Dell support centre. At this point, the laptop's keyboard and mouse refused to work, making it appear as though the system had frozen - the mouse cursor had turned into the busy-in-background animation.
However, as a last ditch try, I plugged the receiver for the wireless mouse that I'd always used with the laptop (even during the first and fatal Windows-based BIOS flash attempt) which I'd removed to make space for the BIOS recovery USB drive, into another USB port and was able to use the wireless mouse to close the "update failed" dialog box. Upon doing so, Windows attempted to restart the laptop. The screen turned black after some time and nothing further happened. The HDD read/write LED remained lit solid and holding the power button down did nothing. I once again pulled the AC adapter out. I kept the USB drive inserted and "drained any residual power" by holding down the power button for about 30 seconds. Then, per the method in your link, I held the End key down and while doing so, inserted the AC adapter. The laptop sprang into life and went back into the BIOS recovery system, where it went on with the flash update - again.
Once finished, the laptop rebooted - this time into Windows, although very slowly. It appears to be working now, I will keep monitoring for stability.
Long story short - the method in your link can work with a Vostro 3700, as long as it is followed exactly.