If he had known she had too many commitments, he would have done something about it. (Backshifting within the scope of a modally remote preterite)

Solution 1:

This is an interesting example of how grammatical analysis and formalism may create a line of reasoning that is more obscure and more complex than the prototype.

The Cambridge account is akin to my describing why we draw our hand back when catching a cricket ball, by using all the specialist terms of energy, force, impact, momentum, friction and interatomic interactions. The simple answer is that it hurts if we don’t.

So it is with this question. We may seek understanding of the subtle combination of the rigorous grammatical concepts or we may seek a simple working view of the two sentence alternatives that you consider. Two rather different problems. I am not competent to undertake the former but view the latter as:

If he knew {at this present time} she had {during the time leading up to and including the present time} too many commitments, he would {at this present time} do something about it.

If he had known {at some previous time} she had had {during the period leading up to and including that time} too many commitments, he would {at that time} have done something about it.

This answer is not to dismiss the importance of grammatical analysis but to distinguish it from the more common pursuit of meaning.