The gap at the side of the neck will have minimal effect on tone, it does mean that a really hard knock will shift the neck sideways, which is why jamming some shims in the side(s) might help. The main force on the neck is pulling straight back towards the bridge, about 150 to 200 pounds pressure depending on string gauges. Flying V's have no neck pocket to speak of, and they work fine.
The tight neck pocket is a sign of precise engineering, that's all. There's a bitty trick for tightening the neck, might as well post it again:
Wen you've got the whole thing together and strung up to pitch, slightly loosen the neck screws - a quarter turn ought to do it. You may hear a satisfying "CLUNK!" as the strings pull the neck tight. Check for alignment side to side, then re-tighten the neck screws.
I might be more concerned about how wide the bridge is - it would seem like that neck pocket is wide for a reason? If the string spacing at the bridge is wider than about 2 3/16", you may have strings falling off both sides of the neck. Now, Warmoth makes a "Superwide" neck at no added cost:
http://www.warmoth.com/Guitar/Necks/Superwide_WarmothPro.aspx
In the second pix, you can see how they undercut it to make it fit a 2 3/16" pocket. You could ask them to not do the undercutting - only they can tell you.
Do you know the history of the body? I mean, was it from a 12-string or something? It's pretty odd for any manufacturer to go "off the reservation" of standard Fender specifications. But, as Mayflower mentioned above, "original Fender Specs" were pretty loose-fitting and they had to be for production reasons. They didn't have "luthiers" meticulously hand-fitting each neck to it's own body, they just had a bin full of necks and another one full of bodies, and every neck had to fit every pocket, or they're losing time. Brad Paisley and country/jazz blowtorch Scotty Anderson made their bones on 1968 Telecasters, and you could drive a truck through the gaps on the sides of those neck pockets.