Yes, you should definitely let the crew know; it shouldn't be a problem particularly if you board together and explain that, despite the names and seat numbers they have on their manifests, that you are going to swap seats from the outset.

Don't draw further attention to yourselves by "swapping" seats midway, or by having the Economy passenger attempt to "visit" the Business passenger. Just do a simple clean swap right from the start, with crew knowledge, and leave it at that.

Enjoy the flight!