Fusion time, baby
Oh boy, here comes the handwavium. For starters, the absolute easiest fuel present in the human body... is hydrogen! Because of what you've said in response to Trish, I'm going to go in the opposite direction and go as far from human biology as I feel I can manage. So we're going to make a magnetically suspended toroid (donut) of hydrogen plasma, and get it hot enough to undergo fusion. Hard to start, surprisingly easy to maintain.
A few things we're going to need right off the bat: hydrogen, carbon, iron, and copper. Hydrogen is actually produced constantly in the body by the krebs cycle in the form of free-floating protons that pass through a corkscrew like protein in the mitochondria to produce ATP. We're gonna borrow some, as we only need a few milligrams of the stuff. Considering that the human body uses 100-150mols per day (if we stole that it would be 100-150g of hydrogen, orders of magnitude more than we need), we're going to have plenty.
Next, we'll need some carbon. This will take some finagling, because what we really need is graphite which as I'm sure you can guess, is not produced naturally in any biological process I know of. However, Hydroxyapatite, otherwise known as bone mineral, is precipitated naturally. So, this is not really so much of a stretch in all honesty.
Iron. Our plasma toroid needs to be magnetically suspended so it can be nice and stable. Believe it or not, the biological precipitation of magnets actually has solid precedent in Magnetotactic bacteria.
Copper, also found in the body although to a much lesser extent (your wizards better eat a lot of mushrooms), can be created in a similar fasion, even easier as it doesn't need to be magnetic.
Design and Operation
Our toroid of plasma will be magnetically suspended (using ferrous and electromagnets) in two layered toroids of graphite. The space between the two layers will be filled with liquid helium which will catch neutron radiation to produce heavier hydrogen atoms (deuterium and tritium) to fuel our fusion even more efficiently, and most importantly, help our graphite stay relatively cool. Assuming the fusion has already started up, we'll be using Neutral beam injection, which uses ionized hydrogen (H+) that the body so kindly provides through the krebs cycle, and shoots it into the plasma to stoke the flames of fusion.
Our design is essentially going to be a scaled down version of the ITER Tokomak, which is where I'll be getting all of the following numbers. Making it about the size of a large donut, you'd have a comparable power output of 254 watts, using half a nanogram of hydrogen per second. Using Fusion power from a fraction of the body's atp production. This would produce your 200 megajoules in 218.7 hours, or a little over nine days.
Ignition is going to require a pinch of handwavium, as with an efficiency ratio of 10, you'll need to provide 25.4 watts of input power to start and maintain the reaction. Considering that an elite cyclist can maintain a power output of 1000-1100 watts, this is honestly pretty easily attainable. The application of that power can likely come in the form of an upscaled mitochondria, which already use electrical potential to produce ATP. This is biologically improbable, but it's nowhere near impossible.
I always hate to say something's improbable, but to produce 200MJ of power instantaneously out of something that small, would not be containable using hard physics. The waste heat alone of just about anything using that amount of energy (like a fission reaction) would likely incinerate your wizard. This is where I'm hoping your magic comes in. The fusion power can easily provide tons of energy from biological sources, but I think the storage itself would have to be made almost entirely of handwavium. But hey, that's why you have magic! Perhaps the energy generated could be stored directly as 'magical potential'.
Regardless, here's an overview of your wizard's energy requirements and performance with this organ:
- A little over nine days to produce the energy
- 10% of that energy must be consumed as calories to maintain fusion, that equates to an extra 531 Kcals per day (How efficient!)
- The storage method must be magical
- Bonus points if someone hits the organ as it could cause a teensy tiny magnetic explosion followed by a plasma flash. Ouch!