Holy smokes, there is a lot going on here, and there are some wires getting a bit crossed.
First, we need to define some terms and establish goals. Then we can move on to cost/benefit analysis.
Green manure, while it can contribute nitrogen, especially if a legume (e.g. clover) is used, is probably mostly used to add organic matter to the soil. In your situation, you have not planted a nitrogen fixer, so your "green manure" would be mostly for organic matter. The answer to whether or not you should disc it in lies mostly in the level of organic matter you already have in your soil in that plot. There is also the issue of nitrogen in the soil being used up by microbes while digesting the high-carbon "green manure". Even though wheat might go into the soil green, and will tend to lose nitrogen as it dries out. Same for anything, really, though more herbaceous plants will have a higher nitrogen to carbon ration, they will still tend to lose a lot of that nitrogen to the atmosphere as they dry.
If you had planted a clover species, there is still the question of whether to till(or disc) it in or just terminate the crop. This also depends mostly on the organic matter in your soil.
Probably, for you, in this exact situation, this is the best thing to do. Because your goal is a successful dove field of sunflowers, I would reckon your best bet is to terminate the plot with glyphosate soonish, and then again a week or two before you plant your sunflowers. You will probably also want to roll or lightly disc in order to knock everything down, even though the cotyledons of the sunflowers tend to have decent energy. You want a thatch that will hold back small seeds by exhausting their energy reserves while allowing larger-seeded sunflowers to break through the thatch and receive sunlight before the energy in their cotyledons runs out.
Additionally, using commercial fertilizer gives you the confidence that the field will be a success.
Near the end of the season you can reassess and evaluate whether you might want to plant a different cover crop, and maybe till it all in, pending a soil test.