I think that the second part explains the first:

We can't run this bar without her. [Of course we can. I meant we can] Not [do both run this bar without her] and raise two kids.

I think that the second part explains the first:

We can't run this bar without her. [Of course we can. I meant we can] Not [do both run this bar without her] and raise two kids.

_{(Other answers have explained how the first part is inserted into the second part. But I think it's important to explain what the outcome is, the why. That, as said, the second explains the first.)}