Can I send a message from a phone number and have it show my business name as well?

No. SMS technology does not allow a message to show both a phone number and a business name as the sender.

An SMS sender can be either a phone number or a business name, known as a custom sender ID. Business names are limited to up to 11 characters and cannot receive replies.

You can, however, have both options available on your account at the same time. Many businesses use a custom sender ID for outbound messages where replies are not required, and a dedicated number when they want customers to be able to reply. You can choose which sender to use each time you send a message.