We can quibble about how to describe it, but members-only unions in the US do not have the same standing as majority unions in the US or members-only unions elsewhere. Employers are free to ignore them. They can't negotiate exclusive benefits. Members can't commit to collective bargaining. The prevailing legal opinion seems to be that the NLRA does not allow a union to represent more than 50% but less than 100% of a potential bargaining unit.
Well, obviously. They can make whatever negotiations that they want for their own benefits.
What they cannot do is negotiate for other people who have chosen to not join the union, which is what the word "exclusive" implies.
So yes, they can negotiate. For themselves. Not for other people. Which makes sense, why the heck should they have the right to negotiate a contract for people that they don't represent?
The can absolutely negotiate for themselves, though. Just like you and I can negotiate a contract for ourselves.
It's the other way around. Employers are generally prohibited from bargaining with minority unions.[1] They can change policies that apply to everyone.