The popular idea in our society that is underpinning this question is that funerals are held to allow the family to grieve, to say goodbye, to achieve “closure.” At most, some might say that they are to “show honor” to the deceased.
This isn’t the Catholic understanding of the purpose of funerals. A funeral is primarily held to offer prayers for the repose of the soul of the deceased. This is most perfectly done in a funeral Mass because the Mass is the most powerful prayer the Church has, and makes Christ himself present at the funeral. No Protestant funeral can offer that, and for that reason alone a Catholic should make clear in his final written instructions that he wants a Catholic funeral Mass, especially if he is the lone Catholic in his family.
If the non-Catholic family wants a non-Catholic memorial service for a Catholic loved one, I see no reason why that cannot be done in addition to a Catholic funeral Mass; but such a service should never replace a funeral Mass for a Catholic.
