A girt is roughly analogous to the Riegel of a German frame. The way a German frame is assembled, you come to the last cavity where the riegels cannot be inserted by means of a normal tenon joint, because you simply can't get them in. This is the same problem you have here with not being able to insert a tenoned girt into already standing posts.

What the Germans do here is they use 2 solutions. 1, the riegel is 'swung in' from the top. Normally this involves an extra long mortise with a slanted housing. The riegel has a regular tenon on both sides. One end is inserted into a mortise on the first post, and then the timber is just dropped down into the other mortise. You can wedge this if you want, but gravity will be sufficient to hold it in place.
The other solution is to turn the mortise so that it runs along the width of the grain, and runs out one side. Thus, the timber can be inserted from the side. This works only with very short tenons. This is OK for German framing where things are small, but I wouldn't recommend it here.
However, the drop in tenon would work rather well I'd think.

DLB