I once recorded a song that had to have the sound of a live audience in the mix. I had the same trouble, I just couldn't get a good sound with just myself as an audience and using multiple overdubs.
 The solution I used was to get 3 friends together and recorded the four of us around two Omni mics. There was enough noise so that when I used multiple overdubs (five or six), it sounded like a huge crowd. This could possibly work with clapping as well, it's just a matter of finding out what is the lowest possible number of people that will give you the difference in sound you're after.
Good luck.

Thanks everyone! I decided to find clapping from a sound library. I am having to manufacture it from multiple instances of single claps, because this is a very smart fake audience, who knows how to clap on downbeats over rapidly shifting time signatures. 
 
As a thank you to all who helped me, I'm offering up this video I found while hunting for my clapping. Harry Connick Jr playing live. The audience is clapping along on 1 & 3, and in the middle of his solo, Connick adds an extra beat to get them to 2 & 4, and you can see the drummer's reaction in the background. It's so cool.
https://www.youtube.com/watch?v=yD3iaURppQw