Can I buy the license for my client, and bill it back to him?

Yes. Once you have purchased your license for your own use, you can buy another license for your client. Go through the chekout process on Myfonts.com, and then fill the “Who is the font for?” form with the name of your client. The bill will be on your name, but the licence on your client’s name. Then you will be able to invoice it to your client.