Once the procession reaches the funeral venue the coffin needs to be taken from the hearse and carried inside.
In a ‘traditional’ funeral the customary order is: first, the priest or celebrant, then the funeral director, then the coffin, then the chief mourners, then everyone else.
If you think it inappropriate for the coffin to be preceded by two relative strangers, decide on a better order—and get rid of the strangers. You won’t be able to do this at a church, where you are the guest of the priest.
Think about what music you would like to be playing as you come in. There is only one right way: your way.