The Klingons have allies of the Federation for a century. They just helped off The Dominion and The Borg. Them joining the Typhon Pact alongside their enemies the Kinshaya and the Romulans just wouldn't make any sense.
Obviously not with the Kinshaya, but I note they have the most to gain. The Empire has never really expressed any desire to join the Federation and their attitudes are incompatible. Besides, they were within inches of going to war with the Federation before the Cardassians became a much easier target.