Okay, I already found it. Turned out, I was editing the wrong .ini file. In case somebody seeks the same answer in the future, the file you want is "DefaultInput.ini" in the "\Alpha Protocol\APGame\Config" folder. If you have the Steam version, this is in "Program Files\Steam\steamapps\common".
In this file, you can add under the line "; --- General bindings" the line
+Bindings=(Name="F4",Command="togglehud")
to make f4 toggle the hud.