Does the jack plate not fit in the socket? Sometimes it helps if you loosen the nut and twist the actual jack connections round on the plate as in some positions the tip connection can touch the body and keep it from fitting correctly. It's worth testing this with a jack plug installed as well, just so you get it so that it doesn't touch.

If you take the jack socket part off completely, does the plate sit in the cavity correctly or does it still sit proud? At worst, you may have to use a file and enlarge the cavity slightly, probably easier to do this at the thinner end (away from the jack input).

If you are going to lacquer the neck, then you will make the neck slightly thicker anyway, so it all depends on exactly how much of a gap there is. As there is a gap, (then although it's not normally done), you can always spray the sides of the neck pocket to add some thickness and reduce the gap. If it's sizeable, then people have got a bit of very thin maple veneer and stuck that on the edge of the pocket.