Would a fretboard recap solve help compensate for neck warping?

If you adjust the truss rod, does the shape change? Does it seem to bow between the nut and the fret where the neck joins the body, or from the nut to the end of the fretboard? If adjusting the truss rod does nothing, does it rattle if you rap on the back of the neck when the rod has been tightened? That would indicate a problem.

The fretboard isn't really a major structural component of the neck.

 

A fretboard can be planed to account for mild uncurable neck issues, but it's really a last resort.

 

Applying a new fretboard, or for that matter removing and reattaching the current one, can help with the problem. If the problem is up bow, for example, regluing the fretboard while the neck is slightly back bowed will help to straighten the neck.

The strength comes from the lamination, not the fretboard itself.

 

You probably wouldn't even need to go that far. "Heat setting" - heating the neck enough to soften the glue joint and letting it cool in a slightly back-bowed position can often fix this issue (assuming there really is an issue - OP, you haven't said whether or not you tried adjusting the truss rod). In any case, you're probably best off bringing it to an experienced luthier and having them assess the neck and give you options for how to deal with it. Don't be afraid to get second opinions, either!

 

If you have another fretboard put on, you can have the neck itself planed and adjusted and it should take care of the problem, as it seems yours is minor. You can also have it PLEKed, which also might take care of it.