Is that gasket material around where the old doors were, sealing that gap well enough? You could be losing draft because it is pulling air from that gasket. Maybe try sealing that with refractory cement instead.
Thank you for the helpful comments, folks. I sealed the hole at the bottom of the chimney, and that seems to have caused a noticeable increase in the draft. We've tried burning it hot today, and haven't had any trouble with smokeback yet. Maybe that will be all we needed to do!
I don't know how to tell whether the superwool is forming a good enough seal in the gap where the doors used to be. I'll try holding a lit match near it to see if there's any air movement toward the seal.
I haven't tried raising the plunger tube yet; as I said above, maybe I won't have to.