Spray bottle for sure. Personally I like a lighter seed starting mix to start, then transplant to castings / manure / more substantial soil.
Also. many people put the tap root down when planting, but concentrate on which direction the seed is heading. Even if it means the tap root points up when you put it in soil (because the seed head and tap root are in a U for example), it takes more energy to change the direction of the future leaves than to redirect the tap root.
I also try to have the helmet just on the surface when I transplant the sprout