Ok, but as an AL DM, what about the staff?
As a (former) AL GM, my understanding of the rules is:
An arcane focus (staff) can be used as an improvised weapon. Since it resembles pretty closely a quarterstaff then it would do 1d6 bl damage and have the versatile (1d8) tag. Same for an arcane focus (rod) and a club.
An arcane focus (staff) is not a magical weapon, so a Pact of the Blade warlock canot transform it into their pact weapon. It is not a weapon, so the same warlock cannot create a pact weapon in its form. Similarly, a Hexblade warlock cannot use the Hex Warrior feature to channel their will through an arcane focus (staff).
If a Pact of the Blade warlock wants to use their pact weapon as a spellcasting focus then they must take the Improved Pact Weapon invocation.
What about magic items? Some GMs allow magic rods and staves to be used as spellcasting focuses but that is a houserule - no rules in the book say that any magic item can be used as a spellcastiong focus. Crawford did, however, tweet a couple of years ago that wands can be used as focuses. You may or may not treat that as an official rule; most AL GMs won't.
Magic rods, staves and wands are not weapons (for example, the description of the rod of absorbtion says "Rod, very rare (requires attunement)", not "Weapon (any)." You can use them to hit someone, using the improvised weapon rules, but they are not weapons for the purpose of features.
In short, an "arcane focus (staff)" is not a "quarterstaff." An "arcane focus (rod)" is not a "club."