It's obvious that Dysley fancies Lightning. So, he called her Claire to fulfil his sexual fantasy. It's either that or he thought calling her Claire would convince her to change her focus. ''Why did he call me Claire? Does he like me or something? Should I feel sympathy for him?''
Also, if Claire was actually her real name, others would go like 'WTF? That's your real name? What's going!?' But they didn't.
This game has so many ... Sexual stuff.