Brian
At one time I was a professional sunflower breeder. I was working on the perennial Helianthus species, but breeding methods are much the same.
Just cross a red one with a giant one. Grow out lots of seeds. They should have yellow flowers. Save seeds again, and grow them out. Some should be red, and bigger than what you started with.
If my memory is wrong, and the red does show up in the first generation, then it will be easier. Just cross red-flowered sunfloewrs back to the large-flowered variety every time you get the red.
Do not use a commercial F1 hybrid sunflower for crossing, at least not as the seed parent. Those commercial hybrids are made using a male-sterile strain as the female parent of the hybrid. Those genes may mess you up later.
So cross the red hybrids with the large headed variety again. Again you will have yellow so grow out another generation from them. Some should be red again.
As soon as you get some red sunflowers that are pretty big, you can stop backcrossing, which is what this system is called, and start selecting in the population for bigger flowered reds.
With 2 years per backcross, it will take a while, but much of the size change will be early, as some of the size is due to the sunflower putting all its energy into one flower head instead of many smaller flower heads. And that is due mostly to one gene.
You can choose to stop backcrossing to the big variety when you want to.
It is important to grow seeds from more than one plant in each generation. Many sunflowers are self-incompatible. That means each plant needs pollen from a different plant to set seeds. As few as 6 plants per generation is risky, because closely related plants may not be recognised as different plants.
If you only have one red hybrid in a generation in which you had planned to backcross to the large variety, no problem. Mix the pollen from several large flowered plants and use it on the one red one, and use the one red flower's pollen on several large flowered plants.
Sunflower's pollen will keep for a few weeks, if kept dry. Water kills it on contact.
Sunflowers have never been successfully crossed with anything except other sunflowers. Crosses between some species of sunflowers are really dificult. Some have never been successful. And yes, they have been tried in all combinations. Dr. Heiser, for one, made crosses in all possible combinations to see which would cross and which wouldn't. The annual domestic species will cross easily with some wild annual species. It will cross with some difficulty with some perennial species, Jerusalem artichoke being one.
Good luck and have fun.
Walter