It had a round ferrite core in line already, and it drew .5a standby so I assume that you're saying it doesn't need the ecore.
As far as grounding goes you may know that the American 240v is made with two hot lines (l1 and l2) that have 120v of potential relative to the grounded current carrying conductor (known as the neutral) end the two hot lines have 240v of potential relative to each other because they are 180 degrees out of phase relative to each other (picture two sine waves laid up one each other to make a string of infinity signs) so our common mains has l1, l2, n and ground (neutral and ground are commonly bonded at one spot in the system) your typical setup internationally is L,N and G (or PE/E) on the power Jack I have there is an output block that has the international L,N,E terminals, an input block that has the L,N,E terminals, and an output block above them that has an L1, L2, N arrangement. The LNE output on the bottom reads 120v between L-N the block above it L1-L2 reads 240v should I just not hook up the N on the top block? I'll attach a picture of the split phase unit's front.
I once wanted to charge the inverter with a 240v split phase generator and I had success by tying l1 into the L terminal of the input block and L2 into the N terminal of the input block
Having trouble with Internet, just look up "splite phase 220v 15000w lf inverter on eBay) they spell split "splite"