Wow! Where I am, early June is when the new culms shoot up rapidly.
Most sources I read say to transplant in spring, or early spring. A few sources (the useful ones) say WHY: don't transplant during or close to the rapid "shooting up" time.
That makes sense since that must be the most stressful time for bamboo.
I think that in my climate (long wet fall and late, mild winter) a fall transplant would have been OK, since they would have so long to settle in before hard frosts.
But probably spring is better, as long as I get them WELL before the new culms started shooting up.
I also want to say thank you for the idea about "mow 'em all down to the ground". Mine is so dense with such skinny culms I can't figure how to wean out the older culms. "Cut them all; God will know his own."