The Bears don't fire coaches after one season and I don't know why you would fire Eberflus anyways. It isn't like he has much to work with on this team lol. Besides, this is a pretty kneejerk reaction to half a season. I'd like Fields to have continuity at OC as well, but there's only so much you can do.
Couldn't you just offer Getsy a promotion to assistant HC along with a hefty pay bump to keep other teams from trying to poach him? That said, it's normally not a very considerate move by an employer to block your talent from vertical career moves (even outside the organization), so you'd obviously want to have a candid discussion with Getsy about his aspirations and his value here. Ultimately if he wants to get a HC position immediately, you'd be better off helping him as a GM than being an obstacle to that. Then you could start planning a replacement now if need be.