I see several suggestions for Sandbergs. They are fantastic instruments and generally lighter than Fenders. The build quality is FAR better than anything Fender does. But if you want an "old school" Fender tone, Sandbergs are a very bad choice. Sandbergs have a much more modern, bright tone. Even the passive models. They don't "sit" in a mix like a classic Fender. Not even close. If you buy a Sandberg and want an old school Fender tone, you will most likely be disappointed, no matter how good Sandbergs are.
I suspect some of the other suggestions in this thread may depart too far from an "old school" Fender tone as well.
If you want a lightweight Fender-style instrument that actually sounds like a classic Fender without knowing the specific weight of each individual bass (some dealers provide this info btw) I don't think you will find any lighter Fender basses than the basswood body MIJ Fender Traditional series. Of course there will always be some variations between individual instruments, but overall they're the lightest Fender series I've ever played. So light in fact that neck dive can become an issue. The ceramic pickup in these MIJ Traditionals isn't the best I've tested though. It has a very high output but is also slightly lacking in clarity compared to a good alnico pickup. It's not horrible, and will certainly give you a more "old school" Fender tone than any Sandberg, but there are definitely better Fender-style pickups out there.
MIJ Fenders also generally have far better QC than MIM/MIA Fenders.