All done! Here are some photos for the next person who wants to fix this, it was an easy enough job to do myself.
Just pull down carefully on each edge of the panel, the metal clips will bend slightly to allow it to come out. Just make sure you don't bend the board too much.
Here's what it looks like with the panel removed:
The metal clips just seem to go straight into slits in the headliner vinyl.
I unpicked the staples using a screwdriver on three sides. Left one side attached, made it a bit easier to keep it all aligned.
Then spread some of this stuff on both the board and the back of the suede. Waited for it to dry, then pushed the suede onto the board carefully - it bonds instantly.
Looking good! No more sag, the suede is hugging the curve of the backing board nicely.
You'll need to use 4mm staples, which are quite shallow - anything bigger is too long and will poke through. It was hard to find these, and they didn't really fit properly in my staple gun, but it kind of worked.
Notice the metal clips are still quite upright - that made it easier to put back into the slots in the car. But then it hangs really loose as you can see:
I don't know what the proper way to install these is, but all I needed was to make the metal clips lie a bit flatter. So I grabbed a spoon while wife wasn't looking and made this hi-tech tool (patent pending):
et voila!