The solution I found is to create a custom BBCode called RTL that then can have a button with the other BBCodes. To do this, navigate to the Posting => BBCodes section of the phpBB3 administration panel and click "Add a new BBCode." Then, put in the following:
Make sure to check display on posting page so users know they can use it.
You may also want to create a LTR but if your board default is LTR then it shouldn't be needed.