Kind of two parts...
Do you think Chip could work as a nickname for Caspian?
If no, what names would work with Chip as a nickname?
Charles is out. Crispin is out now too (that was my favorite option).
First of all, I love Caspian. Such a handsome name, I love Cas- names on boys. As for Chip as a nickname, it's a bit of a stretch if you ask me but that doesn't mean you couldn't use it. People use much more loosely related names all the time, so I think it would be fine.
The first alternative that comes to mind is Christopher, since it has all the letters in Chip and I know Kip/Kit is sometimes used as a nickname for Christopher. So why not Chip? If you like Christopher, I think this could be a really good way to go.
Another thing you could do if you wanted to reinforce the nickname-fitting a little more is have initials CP. I've seen this done similarly with initials JC for the nickname Jace. So I could see any name, say, Cyrus Patrick, with the nickname Chip, and not be that surprised either. Maybe it's a little bit of a stretch, idk, I just think it's a cool idea.