Affect sound? Not likely.
Will it start to bow? Probably. It's not guaranteed, but it's a safe bet that it will.
Have one installed? It'd be expensive, and pretty invasive. I can see 3 ways to do it:
1) The neck is routed from the back in a fixture to ensure consistent depth and straightness. A truss rod is installed, and a fillet is glued in place (like the Fender skunk stripe). The neck is then refinished, and a body modification may be necessary to add truss rod access.
2) The fretboard is removed and the neck is routed from the top, again in a fixture to ensure consistency and straightness. Truss rod access can be made at the head or the heel. If at the heel, the body may need to be modified to add truss rod access. The fretboard is then replaced, and it is quite likely that an entirely new board would have to be used. The neck will then have to be refinished.
3) An entirely new neck could be made.
So it's a labor intensive task regardless of which way you go. I would recommend #3, but I'd just replace the whole bass before I did that. A new neck by a custom builder can easily be more than $1000, and you can get some damn fine whole basses for less than that.