Preheat the oven to 350 degrees F (175 degrees C). Grease an 8-inch square baking pan.

Make the crust and topping: Whisk brown sugar, flour, oats, and cinnamon together in a bowl. Gradually stir in oil until mixture is crumbly. Press 1/2 of the mixture into the prepared pan for the crust; set the rest aside for the topping.

Bake crust in the preheated oven until firm, about 10 minutes. Remove from the oven and let cool slightly; leave the oven on.

Make the filling: Bring serviceberries and water to a simmer in a saucepan. Cook for 10 minutes, then stir in lemon juice. Whisk white sugar and flour together in a small bowl; stir sugar mixture into berries and cook until mixture begins to thicken, about 3 minutes. Remove from the heat and let cool for a few minutes, then pour over crust.

Sprinkle cream cheese pieces evenly over filling. Spread reserved crumb mixture over cream cheese gently press onto filling with a potato masher.

Bake in the preheated oven until crumb topping is golden and filling is bubbly, about 40 minutes.