Balanced BNC cables

The main difference is that the BNC cable will look like a 1 MΩ (or 50 Ω, 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.

In your case, you don't need to worry about he subtleties of scope probes. If you are going to feed high frequencies through coax into a scope, and 1 usec pulses count as such, what you want to do is set the scope input to 50 ohms, use 50 ohm coax, and make sure your test box output amplifiers can drive 100 ohms. I say 100 rather than 50, because you will want to series terminate your output with 50 ohms. That is, put a 50 ohm resistor in series with your test output amplifier and your BNC connector. This will, of course, reduce the amplitude of the signal at the scope by 1/2, but you know this and can compensate. Either that, or use a x2 amplifier in your test box, so that a nominal 5 volt output is actually driven to 10 volts, and gets divided down to 5 volts at the scope. Either way will work.

Doing this will let you drive a scope to much higher frequencies than you are interested in for this application, so it will do you just fine.