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Would mean needing to buy some plasterboard
Had exactly the same issue when I put in two pendent lights. Buying a whole board for a 3" hole seemed excessive. On the way to the hospital I saw a house refurb with a load of plaster in the skip, so nabbed some of that.
My suggestion would be;
1) go for a ride at lunchtime. There must be tonnes of people having work around you.
2) go to a wickes / similar and ask if they've got any spare broken bits.
Crude drawing here. You could use 2 battens - one on either side if you prefer.
The unplastered board should sit below the finished plaster surface provided it's the same thickness board.