BNC cable instead of oscilloscope probe for fixed setup?
The main difference is that the BNC cable will look like a 1 M&Omega; (or 50 &Omega;, depending on how you set the scope) load to your circuit, rather than the higher impedance of the scope probe. This shouldn't make much difference at all as long as the INA193 has a low output impedance.

As long as the cable is short with respect to the wavelength of the highest frequency of interest, you won't need to think about transmission-line effects.

Also, you won't have the 10:1 attenuation of the probe, so adjust the vertical scale appropriately.