..and then, in turn, perhaps get legitimately sued for abuse of process? Or have a public record that you filed a frivolous lawsuit against an employer? I'm not sure that's a particularly good idea.

Before trying a last resort that can get you into trouble, how about a brief chat with an employment attorney? There may be laws in your state that deal with employee rights to his or her employment records, maybe even including time records like this. Perhaps the laws even allow you to make the request through the state labor department. State laws are very different in this respect so there's no way to know what does or does not apply.