One reason you get more noise when you use higher resistor values is that the Thevenin resistance of your divider (270 ohms, with your present attenuator) causes the common-mode gain to be non-zero. Higher resistor values causes higher common-mode gain (or less common mode attenuation, if that makes more sense to you). For example, if you stick with your present values, you need to add about 80 ohms between your "Low Volts" and pin 2 on the diff amp. If you raise the values of your attenuator to 27k and 3k, the value of the CM gain-nulling resistor becomes about 2509 ohms.
I started out to calculate these values, but the maths got to be too much trouble, so simulated the circuit and determined the values by trial and error.
Note that this calculation assumes that the common mode source impedance is zero. If it is radiated noise, this is almost certainly not the case, and even it the noise is conducted, the source impedance might be significant. If you try this, I would recommend you use a pot (connected as a variable resistor).