The problem with pulling the ground wire from your scope is that you create an implicit 50V (us) or 115V (eu) ~1mA AC source with regards to earth, or DUT's with the plug reversed.
This means wherever your attach ground in your circuit, if that circuit is connected to mains, there could run a few mA of AC current trough it from power supply to power supply.
This is what kills boards and screws up your measurement since the frequency can be 50/60Hz or whatever else is used in those power supplies plus noise on the grid.
I hear you thinking, but what if I don't have grounded sockets? Then you use equipotential bonding. This forms automagically when using a grounded terminal strip. All "grounds" are connected, but it isn't "earth". This is a risk in itself, but sometimes it is done. Read some literature about this before you even consider it.Running the scope on battery is the best of the worst methods. But you'd have to keep in mind that all exposed metal is scope ground and can't be touched anymore. Including the USB and GPIB plug and probe cal.
And you still don't have inter channel isolation.
Can you do it? Yes.
Will it work? Maybe, depends on what you measure.
Is it safe? No.
Does a GFCI make it safe? No.
What is safe?
HVP70 And:
Test it!Pull the ground wire. And then take your multimeter in milliamps AC to make the ground connection. What do you see?
This will run through your DUT when you pull ground.