Not sure that "midrange growl" is entirely compatible with short scale, but you never know.
If you can find one, an 80s or 90s SGC Nanyo Bass Collection, ideally an SB320, would be a good choice and still just about within your budget.
It's not like there's any shortage of very nice basses at that price, especially if you include pre-owned, but the SB320 would be my pick.