What kind of bike do I need?
You'll need a bike with low enough gears to get you and your stuff over the mountains, wide enough tires so that you'll have some traction in loose gravel, and a sturdy enough frame to handle the weight of your camping gear. On the first ride we had bikes with tires as skinny as 35mm and as wide as 3". While you may be able to do it with 35mm tires it will be much easier and more enjoyable with tires in the 2" range. These don't have to be mountain bike tires. Touring tires will do fine as long as it doesn't get too muddy (which it shouldn't in early October).
You don't need a mountain bike although a mountain bike would be fine.
A touring bike will work well if you can fit tires wider than the standard 35mm touring tire. You might want to take the fenders off, as these can be a hazard in gravel, and see if you then have room for 2" tires.
What about a gravel bike?
Almost every bike manufacture now has a model, or two or three, it calls a "gravel" bike. Some of these are great. Some are not geared low enough for this kind of ride. Basically these are sturdy road bikes with room for wider tires and disc brakes. If you have one that's great but there are many other bikes that will work on this ride.
An "All-road adventure" bike is another new category that is really targeted at this kind of ride. In general they will have wider tires and lower gearing than a gravel bike. If you have one of these you're ready to go.
Really about the only bike you would NOT want to use is a skinny tire carbon fiber road bike.
How should I carry my stuff?
Because of the rough terrain and steep climbs traditional touring panniers are not the best choice, but they will still work. Bikepacking style soft bags are a better choice for this kind of ride, or a combination of panniers and bikepacking bags. The main thing is to keep the weight down. Less is more.
What is the difference between Bikepacking and Touring?
One one level there is no difference. You load some camping gear on your bike and go. If you've ever done a Loop Tour and carried your own gear you already know the basics. The differences are dictated by the terrain. On a ride like this we'll be carrying our gear over steep grades on rougher surfaces. This means weight is a huge factor and we want to go as light as possible. One way to cut the weight is to forego the traditional racks and panniers and switch to soft bikepacking bags like those from Revelate, Apidura, and many more. There are several advantages to these kinds of bags. For one they are lighter than racks and panniers. But they are also more compact and higher up on the bike to make riding narrow trails, or fording creeks, much easier. Of course they are also smaller, so you won't be able to carry as much, which is a good thing for bikepacking.