Spraying is really great… give it a go. there is one thing you have to be careful of, well… two things. First you must be aware that the “dust” or “spray” from the pistol will go all over the place…. and depending on the kind of oil you are using, it could take a long time. I am not familiar with the drying time on the Salad Bowl finsih, but I could imagine you chose it because you do not want to poison your child with the chemicals and dryers that often are in other oils. Good thought, just make sure you do not get any of the fine particles of the overspray on things you do not want oil on, like things you might one day want to cover varnish.
Second thing to watch out for is that all surfaces that you want to glue up, are taped over with masking tape, whether you wipe it on or spray, its a pain in the butt if your joints do not stick.
good luck
ps, its is also much easier, like you suspected, many times that you get as much as the finishing done before the assembly, especially with things that are hard to do the inbetween sanding (between coats of finish), it all depends on the piece.
-- Nicholas, Journeyman Cabinetmaker, Partenkirchen, Germany