CMRR test methodology and effect of open loop gain
Just short-circuiting the inputs will not be possible. Usually, the small input offset voltage will be amplified by the opamp and will in most cases lead to saturation of the output.

Using a amplifier in buffer configuration reduces gain, so saturation will not happen at the output. However, it will change the output voltage as you sweep the input. Measuring the difference with the input will therefore give you the sum of the input offset voltage, a contribution due to finite gain and your figure of interest: the CMRR. So you have one more term in there. If the differential gain is much much larger than the CMRR then the approximation will be OK. But I don't think this is necessarily the case. The input offset voltage is more or less constant w.r.t. the common mode, so you can compensate for that.

When working with the differential amplifier configuration, you can add some gain to the CMRR, making the measurement more sensitive (if the resistors are matched well enough). The gain can be limited so output saturation due to an input offset voltage will likely not happen. And because the output stays close to constant, the extra term in the previous paragraph (caused by the finite gain) will also be much more limited.