It looks like I'm a bit late to the party. I take it you've already met with them last week.

In the event that any of this information is still relevant—

You don't need to prove anything to them. The letter they sent was a civil demand; no response is required, though it's best to send them a letter explaining that you didn't steal anything, were not in the store that day, and so you will not pay them anything.

If they sue you, then you would need that proof, but it's unlikely that they'll sue someone for shoplifting who was never at the store to begin with; there are penalties for filing a baseless claim like that.

While they may be willing to offer you something for your trouble, you have no legal claim to demand the $30 transportation cost; you went there of your own free will when you didn't have to.