Under $1k? Look for a 5-6 year old used bike.
Rides like a bmx? Nope. If you try to make a mountain bike ride like a 20", you just end up with a really crappy feeling MTB.
This.
However, I think maybe this isnt what you are asking about anyway.
Do you really want a downhill bike? Or just a mountainbike? What are you going to ride on it?
Personally, I started out trying to end up with an MTB that rides more like a BMX but quickly learnt that: A. Thats very hard to achieve, and B. Even if you achieve it it isnt going to be much fun.
What I am now working towards, is an MTB with a very short back end and higher that typical BB, but still with a nice slack head angle. This has it's own downsides. but it does allow you to have a bit more input in how it rides in my experience without being too compromised at speed.
$1000 is definitely a wildly optimistic budget though. You definitely need to look at second hand and you need to find clueless fucks selling cheap
G.