One of the main problems with replacing the toner is the fact that the particles themselves are extremely small and are almost impossible to clean up once spilled. Be sure that you try to remove the powder from the HP toner in an area that can easily be washed.
Next major problem could again be the toner particle size. If the size is different than the one your printer uses, there will be quality issues with the print itself. If the mass is higher, it could happen that less particles get attached to the drum. They (in the worst case) can leak inside the printer and contaminate it. This also has to do with the voltage of corona wire. If it's too low, not enough toner will get attached to the drum.
Another major type of problem could be different fuser settings. The fuser will bake paper at a certain temperature and if the temperature is too low for the toner you're using, it may not stay properly attached.
Yet another type of problem could be the chip in the toner cartridge itself. I'm not 100% sure how it's done in this particular laser printer, but some printers may just stop working when the counter for remaining pages reaches zero even if there's still toner in the cartridge. I think that Brother printers allow override for this, but I'm not 100% sure.
So your best bet would be to actually at first try to get some of the toner out of the waste toner bin and see if you can successfully use it. If the cartridge is not leaking, when you spend the waste toner, you may try to get toner from another printer and use it.
When selecting replacement toner do pay attention to fuser temperature on your printer and on the other printer and try to match it if you can.
Also if you can, try to find some sort of toner refilling or re-manufacturing service and get them to refill the toner. They may even have some generic toner available which could save you from buying entire toner cartridge for another printer. From my experience it's generally bad idea to open a toner cartridge outside of a laboratory specifically designed for that purpose. As I said, the toner will get everywhere and into everything and may cause problems if you inhale it or if you try to use a vacuum-cleaner on it.
Also just because the printer reports that a cartridge if empty, it does not mean that it actually is. I've seen "empty" toner cartridges that still had ~50% of powder compartment full. Expect it to start leaking everywhere as soon as you open it and breathe anywhere near the powder itself.