I'm not familiar with the naming conventions for sugar in the UK so I apologize if I become patronizing.
Granulated sugar is the every-day table sugar here. It's what I grab a spoonful for my cereal and such, and it is the kind used in almost all of the baking I've done. Is caster sugar what you usually have around?
Caster sugar is called "super-fine" sugar in the States. It is finer that granulated sugar but not as fine as powdered sugar (icing sugar for the UK right?).
Honestly, I can't say that I've ever bought any as I usually only see granulated and powdered, but Wikipedia tells me that "Castor sugar can be prepared at home by grinding granulated sugar for a couple of minutes in a food processor."
That being said, if you're just beating it into butter than I would think the caster sugar should be perfectly reasonable for the job. Though I agree with bikeboy that you should go by weight if you use a substitue.
Running around This Site, I found that there are aprrox 7oz(200g) per cup granulated sugar, and 6.5oz(190g) per cup caster sugar. So you would actually need more sugar if using caster sugar, as it is less dense. Obviously not a lot though (200/190=1.053).
Sucanat or turbinado sugar (Sugar in the Raw) would be pretty indistinguishable from blond cane sugar. One of them might actually even be the same thing under a different name.
sucanat
turbinado
Here's a bit more about the subtle differences between sucanat and turbinado. Time article
Here ya go, from Amazon UK evaporated cane juice UK
Also from Amazon UK sucanat UK
And one more turbinado UK
Best Answer
Yes you can, so long as you work by weight not volume. Unrefined cane sugar (or refined sugar with some molasses added back in) would contribute a little to the flavour, but sugar mainly provides sweetness, and in brewed ginger beer, a source of nutrients for the culture.
Caster at the same volume as granulated would be too sweet (the finer crystals mean more will pack into the same space), but at the same weight it would be fine.