I've had a look around images of Suzuki and aftermarket props and it seems nearly all OE Suzuki have the need for that spacer at the nut end... and many/most of the aftermarket ones have a longer internal bush construction so you only need the washer and nut. Also on other fitment aftermarket props I've had they've come with their version of that rear spacer already pressed tight into the end looking like a fixed part but it's actually a separate spacer you can pop out. Could that be the case with your new one?
In any case I'd fit and not worry.
However the prop you think is spun looks decent condition so worth checking and keeping as a spare. You can bench test by putting a long bolt through the splines, tightening up well and clamping the bolt in a bench vice. Then try and spin the prop blades round the hub with your hands.