It is called electrical circuit for a reason. If you don't ground the probe at measurement point, you will have an unshielded loop that consists of your probe cable, trough scope chassis, out through scope power cable, to the socket, trough common ground on socket installation, through the other socket out into the power cable of DUT, to the chassis of DUT and then to DUT circuit, in whatever way circuit is grounded (or isn't) to chassis. You will pretty much measure everything else except what you need.
Being short of money I understand, all too well. But with physics there are no shortcuts. Wishful thinking is not going to help.
You need differential probe for that kind of measurement. Differential probing can be made passively (trafo) or actively (diff amp).
In which case, as Joe said, you can use transformer as balun (balanced /unbalanced transformer).
Device you mentioned will have filters and other electronics in its schematic. I doubt waveform will have resemblance to what is on the power grid.
You need a separate transformer, that you will run at low magnetization, with minimum magnetic distortion.
It also needs to be high bandwidth, basically an output transformer from an valve amplifier with at least audio frequency range.
I suggest you to take a look at Joe's video regarding this topic. He explains what he did, how and what kind of transformer he used.
Or try to save up a bit and get that Miscig probe. It has high bandwidth (50+MHz very accurately, and up to 100MHz) and is well made, and also comes with all kind of accessories. And is a proper probe.
You cannot replace that with just any improvisation. And it is safe and will protect your scope.
One thing about differential probes that is not mentioned enough is this: with multimeter we have two probes that we can connect wherever we want, and we measure voltage across that component.
With single ended probe you cannot do that, you always measure absolute voltage as referenced to common power return of the device (the ground).
With scope diff probe you can measure exactly like you would with the meter. To me it was reason enough to have it. You can look at the waveform across that diode, resistor, capacitor, transistor...
For that reason there are diff probes that are not made for high voltage but also probes to probe fast logic signals, differential busses etc etc.
Regards,