(A) on the basis of avoidance of duplicated facilities alone

"On the basis of avoidance of" -- the repetition of the preposition is wordy and extremely awkward. In addition, a comma should not normally be placed before a preposition ("on").

(B) on the grounds of avoiding duplicated facilities alone

Same problem as (A): no preposition before "on." In addition, answers with gerunds ("avoiding") rather than conjugated verbs tend to be incorrect.

(C) solely in that duplicated facilities should be avoided

What is to be avoided? The *duplication* of facilities (that is, the act of rebuilding duplicates of facilities that already exist), not "duplicated facilities" (that is, the facilities themselves). The use of "in that" rather than a more straightforward transition such as "because" is also somewhat awkward, and the entire construction is not idiomatic.

(D) while the duplication of facilities should be avoided

"While" signals a contradiction between the two clauses, and that is not the case here -- the second clause serves to explain the first clause.

(E) if only because the duplication of facilities should be avoided

This option sets up a logical relationship between the clauses: Why would building large new hospitals in the bistate area constitute a wasteful use of resources? Because the duplication of facilities should be avoided. The phrase "if only" is correctly used to imply that there may be no other reason that building new hospitals would waste resources. Although it may sound odd, it is a perfectly idiomatic transition that express the requisite idea more concisely than "on the grounds of" or "solely in that."

In addition, this version correctly uses the noun "duplication" and provides the clearest, most precise phrasing.