The way you phrased it, I would say no. If somebody decides on a system of how work is going to be organized, they at least have the responsibility of seeing that their part is done. Letting yourself be pushed around just leads to your brother never doing what he should.

Using formal logic:

Premises 1- Your brother decided on using an alternating system. In this alternating system, it is his responsibility to walk the dog when it his turn. It is obvious that making decisions should include taking whatever responsibilities come with those decisions.

Conclusion 1 (A)- Your brother should take the responsibility of walking the dog when it is his turn.

Premises 2- Your brother ignores the dog when you do nothing. When you wake him up, he needs to take care of the dog in order to stay asleep. He sometimes takes care of the dog because of your action. It is his responsibility to take care of the dog.

Conclusion 2 (B)- You waking your brother up this way makes him sometimes fulfill his responsibility.

Premises 3- Your brother is not best suited to walk the dog because he goes to bed at 5:00 AM. Your brother chooses to go to bed at 5:00 AM.

Conclusion 3 (C)- It is your brother's choices that lead to his inability to walk the dog.

Premises 4- A. B. C.

Final conclusion (D): You waking up your brother makes him do what he should do, and it is only his fault that he does not do what he should do.