I have uninstalled the printer software and drivers at the root level.
I have cleaned all vestiges of the software using driver sweeper.
I have uninstalled all references to the computer where they are found in printers and devices.
I have also gone as far as to copy the spool folder from the printer that works, change the name of the spool folder in System32 to spool.old, and pasted the spool folder from the other computer there. Not even sure if that's a viable option, but at this point I'm beyond frazzled.
I have also done a system check (sfc /scannow), which did find and fix errors, but the problem persists.
After reinstalling the HP software I am still having a problem getting it to install properly over the network, I believe this is an HP software issue and not my concern at this point. However, the printer does install the Windows driver and shows is ready to print, the same issue persists.