For $1000 or less, look into Yamaha, Carvin and the MTD CRB. The Carvin PB and PBJ basses are really, really good for the money.
For $1500 or less, the most recent Fender P5 is pretty good - but earlier editions seem to have meh B-strings, and same for a lot of the earlier Fender 5-string J basses.
If you're willing to explore used in that price range, try to find a Fender Roscoe Beck 5. The build quality on these discontinued axes is way beyond regular USA factory, the B strings sound great (I've had a chance to play two of them, both great), and the electronics are passive but smart and flexible - I keep an RB4 as my four-string, even though I hardly play 4, because I like the axe so much.
Lots of folks like the G&L L2500 (and 2000) when in passive mode. Easy to find used for around $1000, and the new ones are out there.