Your budget limits you, but I'd suggest that you consider a good used rifle rather than anything new for that price. IMHO, you just can't buy a good rifle for $400 new. However, a good quality rifle with cosmetic wear but good bore and action, can be had for that much money and will outlast you if taken care of.
Now about "brush" rifles. Some rifles are limited to brush hunting because of their configuration - short range cartridges, poor sights, or poor triggers etc. A standard rifle with 22" barrel is still shorter than most shotguns used in the brush, and you won't hear many serious wing shooters complaining about a 26" barrel on their shotgun.
So my advice is to look at a rifle that you can shoot well from offhand, that fits when thrown up for a quick shot, and has a good trigger and smooth action. That is what you probably really need, not a short barrel, close range only ammunition, or fast repeat shots.