Good morning. Author of the teardown video was actually kind enough to send me a detailed photo of the board, so I am attaching it to this post. Strangely, it shows that the 'blown' resistor is 30k, not 100k as I expected. So I will try to fix that first and see what happens.
Voltage measurement in DCV mode is working OK, I compared the results with other meter. Current - I cannot compare (my other meter has blown fuse in mA range), but it matches my theoretical calculation of the circuit (LED + 1k resistor on 12V - ~12mA).
Will keep you updated on developments!