The EULA states that code under non-copyleft licenses, such as the BSD license, can be combined with UE4 for distribution of a product - but my question relates to creating an open-source game and releasing code which #includes UE4 headers and makes use of UE4 APIs without including any of the UE4 source or headers in the repository. I imagine that this is fine, since there would be no information present which isn’t already in the publically available documentation, but I’d appreciate confimation one way or the other.
Yes, you can release your code (but only your code) under a non-copyleft license like BSD. This code can #include UE4 headers, access UE4 functions and variables, etc.
But you can’t release Epic’s UE4 code, or modifications you’ve made to Epic’s code, under such a license. Such code remains licensed under Epic’s EULA and can only be shared with other teams with a valid license to the same version of Unreal Engine 4.
Good luck!