RAW... Yes? Yes.
I have looked through the staves rules (notably Casting Spells from a Staff and Preparing a Staff), more staves (design) rules (just in case), spellcasting archetypes (and it's APG entry), Activate an Item (only relevant by tangent), and Cast a Spell (the actual action used by Staves)... and probably a few other pages yesterday.
I found no text that suggested that you needed to 'use' an 'appropriate' spellcasting tradition to cast a spell from the staff. Nowhere does it say "use your archetype DC when activating items with the Cast a Spell activation components for spells of this Tradition" nor "cast spells of the Tradition you Prepared the staff with" nor even "as though you had cast it". What we do have are phrases like
- Use your spell attack roll and spell DC when Casting a Spell from a staff. The spell gains the appropriate trait for your magical tradition (arcane, divine, occult, or primal)
Which assumes you only have one, and
- A spellcasting archetype allows you to use scrolls, staves, and wands in the same way that a member of a spellcasting class can, and the basic spellcasting feat counts as having a spellcasting class feature.
Which doesn't mention anything about proficiency or DC when using said implements.
The conclusion is simple. When using a staff to Cast a Spell, you "Use your spell attack roll and spell DC" regardless of what allows you to choose that spell to cast from a staff.
That seems wrong to me. Not even imbalanced, but just wrong to the spirit of the rules. A Wizard using Religion to Trick Magic Item a scroll of Sanctuary doesn't get to use his spellcasting DC. Why would they be able to just by having a Cleric Dedication? I think a common sense ruling is that you use the attack/DC you would have when casting such a spell. This might mean that different spells on a staff are cast with your primary vs archetype proficiency, depending on tradition overlap. /rant