Would it be possible to use FxCop to check private parts of the API too and apply the rules there too

I would like to have the option do this and get at least warnings.

Wouldn't this be a good idea

Re: Visual Studio Code Analysis and Code Metrics Support for checking non-public API too?

Todd King - MSFT

If you are interested in enabling functionality in your own custom rules you can use TargetVisibilities.All to make your rule check the public and private parts of an assembly.

public override TargetVisibilities TargetVisibility
get { return TargetVisibilities.All; }

To enable this for existing rules there is an option in the FxCop standalone under project options on the Spelling & Analysis tab called "Run all overrideable rules against all targets" I think this is the option you are looking for.

-Todd King