(A) he said that it must be posted on the bulletin board so that both management and labor will know what everyone is

Except “what,” I didn’t find anything suspicious in this option. “What” is generally preferred in interrogative sentences. As of now, let’s move on to other options; we will come back.

(B) he said it had to be posted on the bulletin board so that both management and labor knows what everyone is

Unnecessary past perfect tense.

(C) he said that they would have to post the assignments on the bulletin board so that management and labor knew what everyone was

Unnecessary use of “would have," when there is no counter conditional clause.

(D) he said that the schedule would have to be posted on the bulletin board so that both management and labor would know what everyone was

Same as C.

(E) saying that the schedule had to be posted on the bulletin board so that both management and labor would know what everyone had been

Subject missing. Who is “saying”?

Left with only A, which doesn't have any solid grammatical mistake. In fact, after going through all other options, “what” in A perfectly fits in the context.