Here is how you can make a curved part. I just quickly did this so it isn't pretty but you will get the idea. You would make templates not much different than these. Make a template of the shape and then cut it down to the width you need. Transfer it's shape to metal and cut it out of the metal. Then get a piece of flat stock the width you want and bend it to the shape of the curved pieces edge. Keep shaping it until it fits well then weld them together, wham, curved "tubing". You could make actual tubing by making two metal pieces of the curved one and with two pieces of flat stock make bent tubing!
On the tubing you need thick wall tube as it will be less likely to kink. Bend it over a large radius using leverage of a much longer piece slipped over it. Believe it or not you can bend 1/8" wall 1" tube pretty easy if you have enough leverage.
But not knowing how tight of a bend you need to make, it may not be possible as it will still buckle. But it takes a little experiments anyway so get at it.
Remember that where it is common for the wood to touch the back side of the other skin, you don't want your metal to so that as it will likely be seen on the outside after some use, keep it away from the skin a bit and you can weld it or even bolt it with small machine screws. Being you are using 1/8" or so thick metal you can drill and tap holes and hold it to the lip of the metal on the body with machine screws.