I'm not real certain whether the OP was asking for advice on how appropriate it is for a child to sit at the bar. If so, then you've gotten a lot of opinions on that.
As far as whether it is allowed for children to sit at the bar --- I'm not absolutely certain, but I think that it's probably allowed. I know that children are allowed in the bar areas. At Disney, I don't think that there are different rules for sitting AT the bar.
That being said --- Unlike my restaurants here at home, you will not always have the restaurant menu available to you in the bar. The bar menus that I saw were often very limited in food offerings. That might be the case primarily in the resorts where the bar is physically separate from the restaurant. In the parks, the bar is often within the restaurant and probably can get anything from the kitchen.
I would call Disney Dining to inquire whether an ADR is needed. Even so, and even in the middle of the afternoon, you might have to wait for 5 seats to open up at the bar.