Has this been proposed at all?
Is there anything weird about american policing/law that would mean a national, federal agency wouldn't be able to take on this role?
It would be unconstitutional for the Congress to pass laws taking full oversight of state and local police (it's not one of the Congress's enumerated powers, and per the Tenth Amendment 'The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.').
Sometimes police are found to have violated federal civil-rights law, but that is a bit of a sketchy backdoor.
The Right Thing would be for each state to create such a commission, and for the Congress to create one for federal law-enforcement agencies.
Actually, ensuring that states provide equal protection of the laws to all persons, that states do not deprive persons of life, liberty, or property without due process, and that states do not abridge the privileges and immunities of US citizens is an enumerated power of Congress; see the Fourteenth Amendment, particularly sections 1 and 5.
You make a good point. I suppose one might be able to justify such a commission on that basis (although that itself indicates the problem with the post-Civil War order).
It can be done. The DOJ does this on a case-by-case basis. And FBI has jurisdiction to investigate some of these things. Bernie Sanders has suggested requiring all police killings to be investigated.
Most Constitutional rights extend to people against the states. So if your civil rights (due process, etc) have been violated, it's against federal law.
It's problematic to expect relief for any but the worst abuses from FBI or indeed from any part of DoJ. They're all cops, and they're not going to be eager to police other cops. A more effective organization would be completely separate, and probably staffed by attorneys who had never served as prosecutors.
It would be difficult (maybe impossible?) to do given the US's Federal system of dividing powers between State governments and the National government. The Federal Department of Justice investigates, but the only power they have is to sue local police departments in court and make them agree to stop their unlawful behaviour.
The US Department of Justice does some of this. They sue for civil rights violations, often resulting in a "consent decree" where they either take over the police department or place them under a variety of restrictions.
This would be much easier to implement at the state level. Aside from a few specific instances with the DOJ such as the consent decree mentioned elsewhere, the federal government does not have any authority over local police departments such that it can do much of anything.
Sometimes police are found to have violated federal civil-rights law, but that is a bit of a sketchy backdoor.
The Right Thing would be for each state to create such a commission, and for the Congress to create one for federal law-enforcement agencies.