As you know, "that he was to do" will is awkward in current English as a standalone sentence (except in unique situations, such as, "John didn't deposit the money at the bank today." An annoyed person might respond, with some frustration, "That [pause] HE was supposed to do."
In archaic usage, it may very well have been possible and common as a standalone sentence -- I can understand the grammar of it (word order has changed over time) but I have no idea if was indeed in common use at any time. It's not the sort of archaic English that modern native speakers will recognize -- unless that modern native speaker happens to be a specialist in Middle English or early Modern (circa Shakespeare) English who knows that it was in fact acceptable in either of those earlier Englishes.
I'd avoid it, until you hear from a specialist who can confirm :)