Brush gun on a budget. Ruger American Predator in 308.
ruger.com
Put a decent low powered optic on it with 1X to no more than 3X on the lower end and you have as good a brush gun as you could ask for. It is lightweight and compact to maneuver through brush. You can spend more if you like, but any accurate, short barreled bolt rifle in a moderately powerful cartridge is as good as it gets. And there is nothing special about 308, almost any similar cartridge that suits your fancy will work.
Forget rapid fire, it isn't important. Make the 1st shot count. In brush an animal can take 1-2 steps and you'll never see it again. And once mastered a bolt rifle is just as fast for AIMED repeat shots as anything else other than a semi-auto if you ever do need a repeat shot.
There is no such thing as "brush busting" bullets. Any bullet that hits brush will deflect. The key is to shoot through OPENINGS in the brush. Pinpoint accuracy and a flat trajectory are just as important at 40 yards as 400.
Most of the traditional lever guns and the cartridges designed for them are going to be 2-4 MOA guns that will have the bullet 2-3" above, then 2-3" below line of sight from the muzzle out to 150 yards. In an open field that is easy to account for, but when shooting at a deer 40 yards away with a softball sized opening in the brush to thread the bullet through that arched trajectory is more of a handicap than in the open.
Modern cartridges like 450 BM, 350 L, etc. have the same disadvantages. If you live somewhere that limits which cartridges you can use, they are certainly a more efficient choice than shotgun slugs or muzzle loading rifles. But given the option there are better choices.
A 308, or most any similar cartridge that shoots 1/2 to 1 MOA will keep your bullets no more than 1" above, or below your line of sight from the muzzle out to almost 150 yards making those shots through small openings possible.
You don't need much, if any magnification at typical brush hunting ranges. But a decent scope does 2 things. It is legal to shoot 30 minutes prior to sunrise until 30 minutes after sunset in most places. As much as one full hour in a few places. And deer move during those 30 minutes before sunrise and after sunset a lot. In a heavy canopy you can't see a deer without optics at that time. Plus, a scope will allow you to find those small openings to shoot through.