Steam! That the trick! Steaming the stays relaxes the fabric just enough so the reeds slide in easier. It's still snug, but I can slide them all the way to the last 1/4 inch, then guide that last bit in by pushing with my trusty needle nose pliers.
I don't grab the end, just butt the pliers up against it and carefully push, applying pressure with my fingers to guide it in. The reeds are all measured and trimmed so there is just over 1/4 inch space at the edge to allow for binding.
I'm so happy. I'd hit that point that comes in every project where momentum is lost and it seems it will take forever to complete. The reed situation was not helping. I feel like I'm back on track though. Hooray.