Oh, oh, Hunter, I must have forgotten to ask "Lead us not into temptation" this morning (actually the Jewish version is: 'Please do not bring us to be tested or humiliated'). And here I was looking forward to a restful day.
Sadly, it is my view that Prof. Beckwith is wrong. Or, conversely, if he is right it is meaningless.
Let's start from my second point and work back. If God does not have to be in Hell because He does not occupy space, then He is not anywhere else, either, so there is no point to the question.
The premise of the question is as follows. The Scriptural idea of 'His honor fills the earth' (Isaiah 6:3) has been traditionally understood by Jewish and Christian theologians alike to refer to a type of presence that, although ethereal, is designed to be a gracing of Creation in a manner that can be defined in terms of Space. Now there is a sort of theological paradox in this, but it is quite clear that Scripture conveys this concept. Indeed it is only because this is true that it is possible to speak of the immanentization of His presence in more concentrated ways in particularized locations, as in 'And they shall make Me a dwelling-place and I will dwell among them' (Exodus 25:8).
Since this is a reality, it now becomes interesting to ask if indeed this presence exists also in Hell, pace your sister-in-law. To answer that it doesn't because the Divine is beyond Space is a tautology and simply not responsive to the query.
If so, what is in fact the answer? First we must say, as Joseph did, '(only) the Lord has the answers' (Genesis 40:8). On the other hand, to the extent that He has revealed glimpses of His wisdom, we are obligated to make our best effort to fathom, just as Joseph, after that introduction, did in fact provide an answer.
Let us approach this matter in stages. Firstly, why would it be problematic for God's presence to be in Hell? We say that it is everywhere on Earth to some degree, including Jeffrey Dahmer's refrigerator and brothels in Thailand with 11-year-old boys and girls for sale. It is even in the chambers of Judge Greer, whose life's prime ambition seems to be the death by starvation of Terry Schindler Schiavo.
So if indeed Hell is a place on Earth, as implied in many verses about Gehenna being underground, then God's presence would have to be there, barring a drastic reinterpretation of the verse in Isaiah. But so what? That level of presence allows itself to be humiliated by the presence of Evil, that humiliation being redeemed in turn by the ultimate victory of Good. And since you need that ultimate victory to redeem the existence of Evil in God's Creation anyway, it is but a small step to the idea that it palliates the offense committed against His presence as well.
On the other hand, if we take the verses about Gehenna's physical reality to be symbolic, and we posit Hell as a spiritual reality that is not bounded by Space, then perhaps we could leave God out of that reality in a spatial sense. But again, this is not saying much, because if Hell ain't spatial then it ain't special not to have an immanent presence there.