I called HP three times.
Each time I was told that because the printer is out of warranty I would have to pay for out of warranty service. Each time I pointed out that my printer is working fine. I never wanted warranty service, I just want instructions that would allow my fully working HP printer (about 3 years old) to work with legitimately purchased HP ink (for over 120$) in my new country of residence.
During the first all I was told to do a normal factory reset and it would "reset the region". Didn't work.
During the second call I was told I would need to use a "secret" support menu, which I could find thanks to a provided YouTube link, to do a "secure reset". The menu only presented me with a few reset options, none of which were called secure reset, and none of which worked. This prompted me to find an even more "secret" engineering menu (Googled my way to it), which provided an extra OOBE reset (out-of-box experience reset). Still didn't reset the printer region.
During the third call I was finally given the "real" solution, which involved using yet another "secret" menu, this time in the HP Smart Windows application (needed to use a VM for this since I run Linux, but I got it working). From this menu I provided some IDs and numbers to the technician, who gave me some codes back. I punched in the codes, the response codes computed in HP Smart were correct, and when I submitted the request a message said that the printer had accepted the region reset... But it hadn't. After a few hours on the phone and many attempts including resets and power offs and such, the technician eventually told me there was nothing more that they could do, and I would probably have to purchase a new printer.
Today I bought a Brother multi-function colour laser printer. So long HP.