Hi John,
I think they both doesn't sound correct actually. To me, you can be inside or outside of a room, and for example if you are outside of a room, you are automatically outside of a room door as well. I don't know, it sounds just weird to point that that person is on the outside of a room door. 'door' is not really necessary to be mentioned. You can easily get the idea of being him outside of his room door.
If you really need to mention him being outside of his room door, I think you could say;
odasının kapısının dışında – This would be the correct expression for your word. But still I must say it doesn't sound nice. Maybe in literature people use words like those, don't know.
odasının kapısı dışında - gives the meaning of 'everything except his room door'
example;Odasının kapısı dışında her yer temizlendi. - Everywhere has been cleaned except his room door.
Ayakkabılarının bağcığı dışında- except the shoelace of his shoes
Bardağının kulbu dışında - except the holder of his cup
I am only a speaker of turkish language so I recommend you to check it from a grammer book. But I hope my answer helps.
Selamlar!