when I set OUTx to 4095. There's a remaining voltage of 0.1v between the base and the emitter which is sufficent to let the current go through the transistor.
4095 is normally the maximum brightness not the minimum one. You get this inversion because you are using a transistor and that inverts the signal and that is your problem. You are always going to get this with the way the circuit operates, you don't see it in normal operation because it is only one clock pulse away from being full on and you don't notice it is not full on. Here you do notice it is not full off.
I am not sure if there is any way round this with the setup you have.