Overexposing the background is 2 stops is nothing for current negative films.
Also, your child is your subject matter, not some random clutter in the background.
Exposure should always be biased to your subject.
Do you have a lightmeter that does flash?
Place it on your child's face and point it to the camera, and fire the flash.
Look what the meter says and transfer those settings to the camera.
You'll always get excellent results that way.