Someone correct me if I'm wrong, but I believe the relevant question is whether Archie left before the beginning of the second semester at TCU. If he did, then he can enroll here for the second semester (last minute or even a week or so late shouldn't be a problem), sit this second semester, next first semester, and be eligible as soon as first semester grades are posted, which in previous years has usually been right before Mizzou.
TCU's website indicates their semester also begins 1/14, so he should be fine on that end.