I'd be more inclined to use PVA than screws to hold the backing: put a
bit of string through the middle, then loop that around a baton on the
front to hold it until the PVA has set. Or you can just add flanges on
two or more sides rather than filling the whole hole with backing -
those can be held in place with spring clamps until the PVA has set.

PVA can be a problem particularly if dust has accumulated on the backside of a ceiling, the screw method is guaranteed success and you don't have to wait for it to set.

Richard