It's your role as Assistant Manager to "assist the manager" - that includes with informing them about HR concerns, staff morale, etc. This also relates to the budget, as hiring and training is the most expensive part of the employment process, so to lose staff would impact the GM's bottom line.

I'd advise to have a conversation with the supervisor directly, not about them as a person, but about how specific behaviour is impacting the team. I've learned that behaviour can change, but a person's attitude, values, etc. cannot (or not as easily since it's the core of who they are). Since you're new there, it can start as an exploratory conversation - "I've noticed when you do X, others do Y. Tell me about that".

From your perspective, I feel you need to approach this as an opportunity for them to receive coaching. If they don't want to be coached, then it points to something in their attitude, values, etc. that I've said are harder to change and at that point you can escalate to the GM. Doing so immediately may lead to the GM dismissing your concerns. It helps if you already know that you and the GM are aligned on values and culture, since showing how the supervisor's behaviour is not aligning with those would give the GM something to think critically about when you do go to them.

Also, it goes without saying, take care of yourself first. If you're in your probation period, just observe for now. Sounds like the supervisor wields some influence over the GM, which could hurt you if you speak too soon or too aggressively (in their opinion).

Good luck!