Well, first I would say that the likely reason for rounded shoulders is mostly not lats but rear delts, low/middle traps, and rhomboids allowing your shoulders to round over instead of being held back in position.
Second I would say that although the workout you posted isn't necessarily overtraining, I think your set-up will compromise your ability to use heavy weights. If you can do this above workout without being tired you are likely not using enough weight. If you intend to use traditional bodybuilding methods, I would split this up into 2 back workouts--more frequency is usually better (within reason and recovery ability) in any case both for mind-muscle connection and for growth stimulus. Push the weight up. Second option is to simply double the workout you wrote above and do it 2x a week if it's really that easy...or to add sets to your workout.
I would split as follows:
4-6x dumbbell rows
4-6x low rows
4x t-bar rows
3-4x dumbbell shrugs
3x barbell shrugs --- redundant with dumbbell shrugs --> either delete and add work to db shrugs, or use a different exercise. I prefer a different exercise.
I would put a few high rep sets for rear delts such as band pull-aparts and db rear flyes or low trap raises in each sitting to work on MMC.