Am I within my rights to arrange it as a cover for non-profit?
I'm not sure about that, but I do know that Broadway Licensing isn't the right source for your license. They told you you'd have to license the whole show, but they seem to have forgotten to tell you that if you do that you also have to peform the whole show. From their FAQ:
Can I change or cut language in the script?
In order to change or remove any language, you must receive prior permission from Broadway Licensing. If you cannot perform the musical without such changes, you should not hold auditions until such permissions have been granted. Please send an email to info@broadwaylicensing.com with the details of the changes you are interested in making. We will respond as quickly as possible.
It sounds as though whoever you spoke with didn't really know what they were talking about. The folks over at Music Theatre International seem to have a much better handle on the situation. Their website describes the difference between the licensing model for a theatrical production and that for a single song:
Grand Rights vs. Small Rights:
The rights that MTI is able to grant under our contracts with the authors are limited to "Grand Rights." Grand Rights cover the right to present the show, in its entirety, on stage. These rights do not include performance of a single song or medleys ("Small Rights"), videotaping, use of a logo or merchandising. In some instances, MTI has separately negotiated representation of additional rights, but those are administered on a show-by-show basis.
"Small Rights" is a term used to cover performances of individual songs in a concert or cabaret-type setting. There are three major agencies that control these rights in the U.S.: SESAC (Society of European Stage Authors and Composers), ASCAP (American Society of Composers, Author and Publishers) and BMI (Broadcast Music, Inc.) Depending upon the songs that are being performed, licenses may be required from one, two or all three of these organizations for a single presentation. Most schools, churches, restaurants and clubs pay an annual fee to obtain a "blanket license" from these licensing agencies that covers the "small rights" use of all music in their respective catalogues for the year. Note that these blanket licenses DO NOT permit dramatic performances of songs. The songs can be performed only in cabaret style. While it is sometimes difficult to draw a distinction between dramatic and non-dramatic performances, a dramatic performance usually involves using a song to tell a story or as part of a story or plot. A blanket license permits neither the use of dialogue from the show nor sets, costumes and/or choreography that invoke the original show.
In general, you need permission to make an arrangement and you need permission to perform it. Whether you're making money or not may affect the likelihood of being given permission, and it may affect the amount of the license fee, or it may not. Copyright does not exist to prevent you from making money with other people's compositions; it exists to ensure that the composers get to make money -- or decide for themselves not to -- when you perform their music.
Your school may already have a blanket license with for example ASCAP that would cover a public performance of the song. At least now you should have a better idea of where to look for it.
