I've force carbonated
this recipe with some minor modifications to excellent results, though it's a little heavy on overall sugar. If you cut down the lemon and add in some cinnamon and maybe cloves, I think you'll get to where you want to be.
My variation in a 5 gal keg is:
3 gal 100% apple juice
48 oz lemon juice
2 lbs sugar
4 oz ginger root
enough water to top up the keg (roughly 208 oz)
Slice the ginger in thin slices, add to the water in a stock pot and bring to a boil.
Add all the ingredients to the keg including the larger pieces of ginger (be sure to strain out anything that will clog the dip tub, larger pieces should be fine)
Chill and carbonate at about 30psi, I prefer the shaking method so that I know all the sugar is dissolved and well mixed.
So if you follow that, but use maybe half the lemon juice (enough to give some tart, but not so much that it will be lemony), and throw some cloves and cinnamon sticks in with your ginger, I think you'll get a pretty tasty brew. May not be identical to Reed's, but should be close and definitely blows Martinelli's out of the water.
You could also carbonate with yeast, but I'd be sure to use a premiere cuvee and a yeast nutrient with it to avoid any sulfer notes.
I do also have an apple pie recipe in my book, but due to copyright owned by the publisher, I probably can't post it here.