I'm going to throw lots of ideas at a wall and make them stick. But changing cameras because it doesn't look the part is wrong. Use what is right for you.
Start with an introduction, and offer to show previous work. Whenever I make first contact with a model (granted it's online) I always introduce myself and provide examples. My business cards I have each one has a different image from my portfolio on it, it's really useful as I always keep a few on me and allows people to see my previous work without the rigmarol of getting a phone out etc.
Make joke of the fact you have a tiny camera! So mixed with what I said above say something like 'hi I'm Belle-Sophie, don't let the camera fool you but i'm a photographer, can I take your picture please. If you're not sure I'll happily show you some of my previous work!'
Wear a camera backpack, even if it's empty. You could be hiding a massive camera in there.
Make your camera look bigger? Could work. When I had my first SLR I used to fool people by chucking a battery grip on it and attaching a flash (even though most the time it was off).
That said, use a flash! Learn about off camera flash/external flash, have that with you and obvious!
A side note, if they don't want to pose because your camera's tiny, these could be people you may not want to work with.