I thought about something to solve this too. My version would be that only half the students are in a house to begin with. So for example:
Black Eagles: Edelgard, Hubert, Linhardt, Petra
Blue Lions: Dimitri, Dedue, Felix, Mercedes
Golden Deer: Claude, Lorenz, Hilda, Lysithea
In Chapter 1, you are assigned as an assistant professor who gets to choose 4 out of the remaining 12 students. Then for Ch 2, Edelgard and Hubert join your party for a mission, Ch 3 you join Claude, Lorenz, and Hilda for a mission, and Ch 4 you join the Blue Lions for a mission. Chapter 5 is the mock battle in which you choose your house, and the other 8 students you didn't pick in Ch 1 are split between the other two houses. This could be done by given every student a first and second choice for house (these are just examples): Dorothea BE, GD ; Caspar BE, BL ; Ferdinand BE, GD ; Bernadetta BE, BL ; Ingrid BL, BE ; Ashe BL, GD ; Annette BL, GD ; Sylvain BL, BE ; Ignatz GD, BE ; Marianne GD, BE ; Leonie GD, BL ; Raphael GD, BL. From Chapter 5 on you can support with the other students and eventually recruit them minus the retainers like in the actual game.