I would strongly urge you to up your budget a little, as you'll end up with a bike that makes riding these trails far more enjoyable. You'll want to look at a hardtail if this is the budget your working with. You will have a very difficult time finding a bike at a local shop that fits within your budget due to the global bike shortage, so I'd suggest you order online.
If you're an XL,
this bike is shockingly in stock and is considered by many to be one of the best values in mountain biking. Has a tapered head tube, which will make it possible to upgrade the fork should you choose to do so later. It has modern boost spacing with thru axles as well, which is basically the new standard.
If you are M or L
this bike is also shockingly in stock. The components are not quite as nice, and you miss out on the thru-axles for QR, but it's also considered one of the best value mountain bikes out there.
These bikes are out of stock 99% of the time and sell out usually within a day. I was considering one myself when I bought my mountain bike, but didn't realize until the following day that I got an in-stock notification in my email, and they were gone.
Factor in spending an extra $100-200 to get a dropper post if you're going to be riding single track. It allows you to drop your seatpost out of the way on the descents to give you better control of the bike. It's a game changer.