Luke brought Louie home to be buried in Stars Hollow, next to his brother William, and this scene is set in Stars Hollow cemetery (it may be behind the church, and only seems to be a short walk from the centre of town, which is unusual for a cemetery, although not unheard of).
The minister conducting the ceremony is identified as Reverend Hackett in the credits (played by Brian McDonald). I’m not sure whether he and Reverend Nichols are both ministers in Stars Hollow, but of different religions, or if Rev. Hackett is a junior minister of some kind. Nor can I tell you if Rev. Hackett is the church minister who plays handball, although he looks young and fit enough for that to be believable. It’s all quite confusing.