A problem that does not touch me as a non-Catholic and non-Irish.
I always thought that Danny Boy is played as a remembrance of an Irish origin and coherence, and since a lot (most of?) Irishmen are Catholic the song naturally is played at Catholic burials.
It surely is no part of a service. But many people (over here, too) like their favourite song performed at their burial as a last being together with their surviving friends joined in a common beloved song.
So why can't you have performed a secualar song at the grave when the service is ended with the blessing issued?