Seems to me that someone concerned about the backstop becoming permanent would not support a six- or nine-month Article 50 extension. If two years is not enough time to come to an agreement on the trade relationship between the EU and the UK, how is six months going to be enough?

Because it is the general agreement that is the sticking point, not the specific wording of the treaties. If general agreement can be reached in six months, then 18 months should be sufficient to get the treaties written up.