You'd be much better off basing your billing rate on the
value that you're providing to the customer, rather than the cost/time that it takes you. This is a very basic business skill, but it's not always apparent to those of us who just like to build things, whether we get paid or not.
Try to get comfortable with the idea that you deserve to be paid very well for your work. (It's something I've had to do.) In my case, it was very uncomfortable to send that first message with my new higher rate. But I spelled out the value of the work I was proposing (which had obvious benefit for the client, but also bolstered my own mental fortitude), and the client happily agreed. And now the client is "educated" as to what this type of thing is actually worth. :)
Good luck!