You don't want to undercut to ensure yourself a job or a project. Charge what you're worth! You can always negotiate, but ensure you're making a fair wage. Good luck in the future!

Review your scope(ish) and send an outrageous bid!

Are there any reviews they can read? Can you sign up to an organisation to guarantee your authenticity?

charge them the time you spent on the project

I'd say you have a case of estoppel. This fuckhead asked you to do something after you had quoted a price. He knowingly allowed you to carry on to your detriment. Estoppel is where you don't have a contract but the situation basically prevents the other person from saying that you don't. If proven, you get real damages as compensation.

There was a pallet of bricks at my local hardware store that wasn't selling because they were like two dollars less than the bricks next to it, so the manager raised the price to be just a couple cents more than the other bricks and they started selling. Like others have said, low price can foster the perception of low quality.

Or they scammed you for the plans.

As a business owner, I understand your dilemma entirely, I used to price myself too low also and didn't realize that not only was I losing money on each transaction but I was losing potential clients because of underpricing. I raised my prices and am in the black for the first time since the collapse of the economy, and busier than ever.