So...for all you know the drivers are correct and the printer works...or not...
...wow. Somebody has to walk you through finding the code file that programs the HP's startup sequence. Then set something like 'Get Passoerd'=False. Or something to disable it.
Wait for a pro--not me.
Conversely, maybe there is a way to change the HP password.
I hope someone shepards you into the gut of the situation deep in an HP file, I'm guessing.
And lastly there's this: https://duckduckgo.com/?t=ffnt&q=disable+HP+password+prompt+linux&ia=web
2nd or 3rd link suggests a BIOS move.
Good luck.