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That has also been my understanding, that the priest must commune at every Mass he celebrates or the Sacrifice is not completed and Mass is invalid.

What an odd thing.

There would be no question as to validity, only liceity (and then, only as regards the priest).

Just to throw it out there, might the priest be allowed merely to simulate reception of the sacrament, so as not to scandalize the faithful?  If the Mass is being celebrated ad orientem, nobody but the altar servers would see, and perhaps not even then.  Having served the TLM many times, I can tell you that you've got so much up there to think about, what to do and where to be when, that if the priest would exercise a little sleight of hand, and only appear to be consuming, you'd have to be mighty observant to notice.

But then there would be the Precious Blood, Which in the TLM is consumed by nobody but the priest.  Perhaps he could have another priest to consume It after Mass?  (But then the question that would beg to be answered, is why that same priest couldn't have heard his confession immediately before Mass.)

Just thinking out loud at this point.  I do have to wonder whether John XII reasoned that it would be a lesser sin not to receive, and thereby violate the rubrics and partially vitiate the Sacrifice, than to receive unworthily.  (I'm assuming that not celebrating wasn't an option.)