You can save the the original cornice but you will have to lift the boards
You have to do most of this salvage work from above:
Vacuum all the dust & debris from above the cornice you want to save
Reinforce the cornice with layers of plasterers scrim (I use jute) and
plenty of poured-in plaster of paris (you can also insert through the
cornice, from below, plated screws which will allow a mechanical fix -
although I'd only do this if the cornice is large). Bring the scrim up the
wall so the the cornice is supported. Leave to harden.
From below cut the offending lath & plaster section away - the cornice will
stay fixed and thus saved.
Replace knackered joist, maybe hanging the new, treated, one on joist
hangers if you can manage to hump it into existing holes.
Make good with section on p/board.