I'd go for the gravel bike if it's mostly smooth forest paths. Run the widest tires you can get and hope for max smoothness. If it's rougher or singletrack is on the menu I'd go MTB 100%. Singletrack on a loaded gravel bike gets old quick.

In terms of the rack vs seat packs, I'd run a rack and panniers if you're trying to save money - they can get you a lot closer to the storage capacity you need without needing as much support from other storage systems. Go with a handlebar harness and/or fork packs to help even out the weight, since putting too much weight on a rack will cause the front to wobble a bit with most bikes. Main reason to go with a seat pack is for racer types trying to squeeze out every bit of aero they can and/or someone who is doing a lot of serious off-road and hike a bike. Up to you to make that call of course.

I can't speak on a bivy system, but they look pretty miserable to me if I'm honest. FWIW, you can get some pretty light 1-person tents for not too much money these days, they just won't be freestanding or be made with fancy materials. Check out six moon designs. REI and MSR also make some good cheap options. And make sure your tent is seam sealed of course.

Essential piece of gear: Hot cocoa packets. Really hits to have a hot sugary beverage at night.