Here's a suggestion:

If your brother wants to take control of all of the caregiving, then let him!

While he is running around like crazy, this would give you an opportunity to spend what quality time there is left with your father, even if you have never had a close relationship.

It is also an opportune time to talk with your father about things that you have always wanted to say to him. Even if he is not in a coherent state, being with him and talking to him will provide him with what he needs most - comfort!!!

However, if you and your brother are running around trying to prove who is taking better care, then both of you will miss out on one of the most important moments of your life, including your father's! Do you really think your father cares at this point who is doing what?

Your father's impending death is really not the time to compete, nor is it a time to think about yourselves.

I think you are both acting childish and selfish.

I know you already mentioned this, but maybe, just maybe, if you hear it from someone else, it might shake you out of this "infantile" behaviour.