I can't see any priest objecting to an appropriately performed song at a funeral. Suspect the complainer had something inappropriate in mind, e.g., a garish or flamboyant act that might be ok elsewhere, but not in a church - and not at a funeral.
The complainer was probably free to ask that the relative's funeral be held somewhere other than a church. And to have a secular person preside. But perhaps his other relatives wanted a traditional service. And they have rights too.
I don't mean this as a personal dig --- not at all --- but are you a Catholic?
I ask because this happens all the time at Catholic funerals. Somebody wants a clearly non-liturgical song at the funeral Mass (say, "Danny Boy" or "Thanks for the Memories" ) and the priest legitimately says, "No, this is a Liturgy of the Church. It is not something designed by me, or you."
And then the priest should helpfully suggest--- as this one did --- that they do the requested song at the funeral home, the parish hall, at the cemetery or at the post-funeral luncheon. These are the appropriate non-Liturgical venues.
This can be explained and arranged graciously. What people do not understand is that the Liturgy is not the "property" of ether the priest of the family. It can't be totally customized according to preference. It is a solemn worship-centered act of the Church.
Or at least it is supposed to be!