I have similar pieces, and I have worked on them occasionally.
This is netted work, or web work—not woven.
My experience has been that it's actually easier to perform a do-over than to try to repair a piece. These tiny beads will usually take one strand running through them, and not two (due to using a new thread to replace missing beads and/or broken thread.
To do this work I use very find silk thread, doubled. And it is placed onto an extremely fine twised wire needle that I make myself. These will run through any tiny bead ever made (based on my experience).
If you need to have a couple of these I will send them to you. Just contact me privately.
Good luck. Jamey