Yes, of course you need an external hardware flasher, because it is a laptop. Laptops have EC embedded controller, which interferes with "software" internal flashing and makes it either fail completely, or write a corrupted BIOS image (which would result in computer not booting next time). That is why in flashrom, a flag for internal flashing on laptops is: -p internal:laptop=force_I_want_a_brick . See more information here - https://www.flashrom.org/Laptops
About external hardware flashing: this method is described in great detail here - http://dangerousprototypes.com/docs/Flashing_a_BIOS_chip_with_Bus_Pirate although most of the time it talks about Bus Pirate programmer, this method is almost the same for CH341A - which is the cheapest hardware programmer supported by flashrom (costs just $2-$3). Just a slightly different flashrom command - mentioned at the end of this article. It will be great if you could reproduce this method - not just for the sake of reflashing a BIOS of your laptop to remove ME, but also you will be able to reflash other laptops who failed a BIOS update and now not booting, - probably earning some good money on it