It is possible to add a delay to the Elgato. Problem is, the delay needs to be added to whatever is arriving first; the Elgato will always be second, due to the inherent ~1300-2000ms processing time for on-device encoding so the video will fit over USB 2.0. The only thing adding delay to the Elgato will do is make it fall further behind the on-system captures.
So for everything to match up to the video signal being sent by the Elgato, everything on-system needs to be delayed by that amount. Unfortunately, Game and Window Captures can't be, so their video will always be 1300-2000ms 'ahead', with action on-screen happening before the corresponding audio, assuming the audio is synced up with the Elgato. Same with your cam; if calibrated to the Elgato delay as a reference, you'd be reacting to things a good second or two after the stream sees them happen.
So yeah, a separate Elgato profile, and on-system gaming profile is a good idea as far as mitigating and managing the inconvenience. Unfortunately, it also doesn't allow you the flexibility to switch between the two if you want to take a break for a bit of console gaming in the middle of an on-PC session.