If I went Monk with the Mauler Archetype and fought with a Bo Staff but went Mountain Stance could I use the Bo staff to do all those good monk stuff with it like flurry of blows and stunning fist sense the Bo staff has the Monk trait or would I need to take the Monastic Weaponry feat to do that stuff with it? And if your going to point out the Staff Acrobat Archetype I'm aware of it but the group I'm in play on fantasy grounds and our GM has chosen to do Agents of Edgewatch and has skipped the Extinction Curse so he hasn't bought it and I don't know if he will just so I could do the Staff Acrobat.
Monastic Weaponry would be needed in order to use Monk attacks like Flurry with a Bo Staff. With just Mauler Archetype you would have proficiency with the weapon, but couldn't do monk things.
Also, you can certainly use Mountan Stance and a Bo Staff on the same character - but not at the same time. Mountain Stance is one of them that has a restriction on what attacks you can use. You can only use the Falling Stone unarmed attack.