A new hair-growth drug is being sold for three times the price, per milligram,
as the drug’s maker charges for another product with the same active ingredient.
Option Elimination -
(A) as - "as" beautifully introduced a clause that is part of the confusion, but the idea here is not to compare the "act of charging" with "is being sold." On the contrary, we are comparing the price. We need a relative clause introduced by "that" to add more information about "the price." It says "three times the price." Three times in comparison to what? Ultimately, we are trying to say this: "A new hair-growth drug is being sold for three times the price, per milligram, compared to the price the drug’s maker charges for another product with the same active ingredient."
(B) than - no comparison marker to trigger "than."
(C) that - ok. it explains "the price." "that" introduces a relative clause that modifies "THE price."
(D) of what - The big problem here is that there is nothing grammatically wrong with this. E.g., The online store charges three times the price of what the local store offers for the same product. BUT a preferred construction is "The online store charges three times the price that the local store offers for the same product." Reason? Clarity and conciseness. Between C and D, C is preferred. I know this is not black vs white, but this is sufficient to eliminate D when we have 100% correct option D and somewhat okay option D.
(E) at which - "at which" can refer to a location. E.g., The party was held at John's house, at which we had a great time. That is not the intended purpose here. Moreover, even if "which" refers to "price" - price at price ....? Doesn't make sense