I think the RemotePI board is not possible in combination to the Argon ONE case.
The problem is, that both can cut the power to the RPi4. Just think, that you connect USB Power to the RemotePi board. If it is off, there is no problem, BUT if it powers the RPi4, power will also go to the Argon ONE case daughter board, since it powers the RPi4 over the same GPIO pins.
And now the Argon ONE daughter board gets power from the wrong side (not from USB connector, but from RPI4). This may destroy some logic on the Argon ONE side...
I would try to solder the IR receivers in the Ar gon ONE case and test it (you have to press the button to power the RPi4 up), but then you dan control it with any remote IR controller. The LIRC article was using the IR receiver installed on the Argin ONE daughter board.