Less than $1K to hold 355 is no problem at all. Any mountain bike by a 'real' bike company (Trek, Specialized, Cannondale, etc etc including smaller houses like Surly and many more) will do that, they are made for jumps and drops and big hits, which magnifies the weight of regular-sized riders.
I am a Surly fanboy, a Wednesday would be a great bike, but if you're just tooling around with your kids it's way more than you need. However once you get into
#2 and getting exercise, you do probably want to push it harder, it all depends whether you just want to get your exercise by riding faster on flat easy trails (in which case Wed. is overkill) or by riding more technical trails.
My personal mtb is a
Krampus -- which I recently bought used, for a little under $1000 including shipping from MN to CA. I found it from the Surly/Salsa group on facebook. If you are interested in Surlys particularly I would recommend you take a look there what the used market looks like.
I'm a big fan of the Surly philosophy evident in Wednesday, Krampus, ECR, Karate Monkey, etc, of an all-steel frame (including rigid fork), and putting all the suspension into the high-volume tires. Especially for us big guys, because with an air fork or rear suspension, you have to pump them up so high for our weight, what's the point anymore? Also 1x drivetrains are a great simplifier now that superwide cassettes are available.
However, there's no reason that 355 lbs can only be served by a Surly or the like. Starting from a budget of ~1000, take a look at what's available from the big brands in the $700-800 dollar range, and anything in there will get the job done. Look for bikes with wider tires (here 2.2" I would consider 'narrow', you want 2.4 at least, 2.6 and 2.8 are getting more common nowadays).
Don't bother with a rear suspension, just look at 'hardtails'. On a mtb the fork is the canary in the coalmine that gives an indication of the quality of all the rest of the components. General rule of thumb is that Suntour forks are junk (and indicate the whole bike is of lesser component quality), rockshox are better, and fox is even better. In your price range probably not many 'great' forks.
Hydraulic disc brakes are a nice plus vs mechanical (cable-actuated), especially Shimano. Sram hydraulics have had problems in the past, but have gotten somewhat better. Tektro is ok.
Also take a tour of your local craigslist, drop us some links of bikes that look interesting to you, we can let you know what we think of them
Happy hunting!