Some general comments on your request:
You're probably looking at around $2500. To me, that's the minimum price for a custom hand made Fender-style bass. Standard woods, good quality hardware, good quality aftermarket pickup, passive pickup system. Fancy woods, fancy finish, fancy electronics add to the price. I'd be suspicious of anyone quoting you much less than that. They would be using kit parts, or working for really cheap.
Making a light weight sub-7 lb. P-bass isn't difficult, but it requires some work beyond what the manufacturers (like Fender) are willing to do. Mainly, the neck needs to be made from a lighter weight wood than maple, with a little extra care in its structure and reinforcement. The body also needs to be made from a fairly light wood, with extensive internal chambering. Some attention to details, and it will be under 7 lbs, balanced, and fine-sounding.
My buddy Jeremy Kirsch (who goes by the name of Freekmagnet over in Luthier's Corner) has recently built some cool basses from cherry and really likes it. Cherry is strong, fairly light weight, in good supply, and finishes nicely. Cherry would be a good candidate for your light weight 5-string. Cherry for the neck and body, a simple transparent finish. You can see some of Jeremy's work on his threads, and PM him to see if he's interested in taking on your project.