I'm not a Baptist or a Catholic, but I know something of their theology. The biggest difference is that for Catholics you can only get right with God through the church. Protestents believe in a direct relationship with Jesus. So it would be appropriate for a Catholic not to do the funeral, but the Baptist to do so.
That is correct and but just one addition -- there is nothing to prevent a Catholic from asking God directly or getting the intercession of heaven. So a Catholic can go through the Church, but a Catholic can also ask God as well.
The Catholic Devotion to the Divine Mercy is an example of using Jesus as a way to get God to be merciful on us. This does not require the intervention of the Church.
Pope John Paul II believed in the Devotion to the Divine Mercy.
The big difference here is that we believe the Church is the Body of Christ: He is the Head, and we are the members. The idea of choosing between Christ and the Church is, on thus understanding, based on a false dichotomy. Christ established a Church. He wants everyone to join it, because it means joining Him.
The same truth is expressed in Our Lord's parable of the Vine and the Branches. You can hardly say you have a "direct relationship" with Jesus and at the same time refuse (or neglect) to be grafted into a life-giving relationship with the rootstock, the vine trunk, and the other branches.