What can I do with batter that has lumpy sugar in it?
Don't most cake recipes require mixing dry ingredients first? Well, what's been done has been done.

The standard professional chef way of getting lumps out of anything is to pass it through a fine strainer. In your case a standard metal pasta strainer should work. Put the strainer over a bowl, pour the batter in and use the back of a large spoon or ladle to gentle press the batter through. The lumps of sugar will be left over and you can just crush them then.

Whipping the batter might work the gluten proteins in the flour which will make for a tough chewy texture. Some cakes actually take this into account and others it is bad, so be careful.