I am planning to take a new portable oxygen Inogen 3 to my father in Romania.
Can I carry it in my hand luggage on the plane?
FAA lists the approved devices and the lnogen One G3 is on the list.
The TSA is fine with them too, even if the passenger cannot disconnect (which is not the OP's case but worths mentioning).
Since I am near 100% you will need to transfer in the EU, here's what the EASA has to say on this:
Portable air concentrators (POC) do not contain oxygen as such; they only concentrate the oxygen in the surrounding area. Therefore they should not be confused with oxygen bottles/cylinders. Under the European regulations, POCs do not have to be approved to be carried and used on board.