A further update to my situation. Last night, my employer got a confirmation from the online Employer Checking Service that I am indeed here legally and I can work in the UK without restrictions. Of course, as I said before, the check merely validated that I did not forge my own Certificate of Application.
Of course, as I always go on about, if the Home Office actually did their job properly, the situation would never have gotten to this point as none of these checks are actually pertinent to my situation. :-/ But if it makes my employer feel better and placates certain people at the Home Office, thereby making my life easier, then fine.