I understand Sythe is a none selective herbicide that will kill grass as well as weeds. What would have been the latest date this spring it could have safely been applied to my St. Augustine?

I don't know what Scythe is but if it is non selective, that means it will kill everything.

You can apply it any time as long as you want to kill the St Augustine along with the weeds. Otherwise there is no time when you can apply a non selective herbicide on St Augustine. Even in the winter dormancy, I believe it is alive enough to die from a herbicide. Bermuda on the other hand, goes completely dormant and can be sprayed before it starts to awaken.

As to picking a specific date, that would vary every year. Grass doesn't watch the calendar.