To send templates from a campaign, there must be an email account with access to the project you're working in.
To do this, go to the Email Accounts page. Click on 'Settings' to the right of the email account, and choose 'Security'.
Scroll down to the bottom of the pop-up window. You will now be able to select each Project that you want to give this email account access to. Make sure to click 'Okay' to save these settings!
Here is a walkthrough of the above steps: