I ran into this problem last night when I went out to dinner with my boyfriend and his family.
There were 6 of us all together, no extra chairs, and my purse was too large to put it on the table (if it's a casual place and there's enough room, I see nothing wrong with doing so). I tried to hang the straps off the chair, but it kept slipping off so I just set it on the floor. Not really the best solution, but we weren't there too long.
My advice? Put it your lap if you can. But if you're really that stressed, just bring a different purse. You don't want to miss out on a fun dinner by worrying.