Cutting the ground wire in an isolation transformer?

I bought a non-tech isolation transformer (500w), which I planned to cut the earth mains ground from the output, but leave the chassis grounded; however, when I opened it up I found the output outlets were grounded to the chassis and the chassis connected to the mains ground. So I just cut the ground. Now my isolation transformer is floating both on the primary and secondary coils. Naturally this means the outer case is no longer grounded and if there was a short within the transformer, touching the case would be bad. Otherwise, does anyone see if this will lead to a dangerous scenario when I am probing the DUT with a grounded oscilloscope? Assume that I plan to connect the black alligator clip of the oscilloscope to any arbitrary point in the DUT.