"A Quiet Music" sounds like an excellent meditation piece for church services. I can understand why you would want to play it in its entirety.
Many times I'm able to find either a different tuning or semi-suitable alternate notes for songs that have notes that require frets that don't exist on my dulcimer(s). There is one song that I really like but the only solution is a 3.5 fret needs to be added. So, this is kinda' off the wall, but I'll offer it as a suggestion anyway. I took a finish nail and filed it flat on one side until it was approximately the same height as the frets. I used Scotch tape to attach to the fretboard between the 3rd and 4th frets (see pictures below). I always play this particular song with a noter and the temporary fret works fine. I wouldn't expect the fret to work repeatedly on multiple songs in a jam session, but it works fine for the song I play and it is used once for each verse.
Flexifrets might also be an option if you're willing to modify the fretboard on your dulcimer (see link to Bear Meadow).