An Arcane Focus can be a staff, but that doesn't mean every staff is an arcane focus. RAW only the specific item listed as "Arcane Focus" in the PHB equipment list is actually an arcane focus, that and a few other items that are specifically listed as being a focus, such as the bladelock's Pact Blade. I couldn't find any rule saying that a magic staff works as an arcane focus, enlighten me if you know of such a rule.
What an inane level of nit picking. I already said I wouldn't let this work, why are you choosing my posts to jump on? But, since you insist:
If you want to pretend a "Staff of Power" that specifically gives a bonus to spell attack rolls and is described as a "staff that can be wielded as a magic quarterstaff" isn't a "specially constructed staff" that could be used as an arcane spellcasting focus, more power to you. I think it's moronic, but go ahead.
But even then, a Rod specifically is a spellcasting focus, and the other item in this scenario is a "Rod of the Pact Keeper".
Are you also going to try and claim that the "Rod of the Pact Keeper" isn't actually a spellcasting focus?