The board recommended
that additional funding for the completion of the town pool, which could be open as early as next year, should be obtained through private donations not through tax revenue. A) that additional funding for the completion of the town pool, which could be open as early as next year,
should be obtained through private donations not through tax revenue.
Wrong. "recommend" + "should" ==> redundant.
B) that additional funding for the town pool’s
completion, which could be open as early as next year, be obtained through private donations not through tax revenue.
Wrong. Modifier error.
C) that additional funding for the completion of the town pool, perhaps
open as early as next year, be obtained through private donations not through tax revenue.
Wrong. Grammar error. --> should be "opened", not "open"
D) that additional funding for the completion of the town pool, which could be open as early as next year,
be obtained through private donations not tax revenue.
Correct. Subjunctive mood. Note: the second preposition "through" [through tax revenue] can be omitted.
E) that additional funding for the completion of the town pool, which could be open as early as next year,
is obtained through private donations not through tax revenue.
Wrong. Subjunctive --> must use bare verb "be", not "is".
Hope it helps.