The legitimate users of my site occasionally hammer the server with API requests that cause undesirable results. I want to institute a limit of no more than say one API call
You say that "all thos extra overhead on EVERY request would be defeating the purpose", but I'm not sure that's correct. Isn't the purpose to prevent hammering of your server? This is probably the way I would implement it, as it really only requires a quick read/write. You could even farm out the API server checks to a different DB/disk if you were worried about the performance.
However, if you want alternatives, you should check out mod_cband, a third-party apache module designed to assist in bandwidth throttling. Despite being primarily for bandwidth limiting, it can throttle based on requests-per-second as well. I've never used it, so I'm not sure what kind of results you'd get. There was another module called mod-throttle as well, but that project appears to be closed now, and was never released for anything above the Apache 1.3 series.