Good question, as I can think of numerous good budget short scales (having switched to short scale myself, as it fits my playing style better than long scale), but very few within that price range. However, all 3 of my short scales did roughly fall within that range!
My main bass is a fretless Harley Benton Beatbass, that cost me around $180 when I got it ($200 with a gig bag... but they may have increased the prices since then) (it's a replica of Paul McCartney's classic Höfner violin bass, and naturally they have a fretted version too). I wasn't expecting such a cheap bass to be so great, but it is! I didn't need to change a thing, it was all perfect, right down to the stock strings! It's also extremely lightweight, and maybe the only issue is neck dive (but that's easy to solve). You can also browse Thomann's website, Harley Benton makes a wide range of budget instruments, and often they are decent quality.
My other two shorties are both second-hand (one is a Jolana Kolorbas, a Czechoslovakian antique from the '60s resembling a 335-style guitar, the other is an acoustic short scale Cort). So I could also suggest looking for one second-hand, I got at least half my instruments that way!
So yeah, my suggestions would be browsing Thomann's website for Harley Benton basses, or looking for a used short scale! I've also heard great things about Ibanez, Gretsch, and Squier short scales, which also tend to be cheap.