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It's not the *wire* that has the characteristic impedance, it's the wire and the ground return circuit that provides the characteristic impedance.
What you have there with your BOG, is effectively an extremely leaky transmission line feeder that happens to have a useful directional response to skywave signals (both front and back along its length if the far end isn't terminated with a dummy load matching its characteristic impedance).
If you can set up a variable dummy load which impedance can automatically track the ever changing characteristic impedance of your 'leaky feeder' antenna[1], apart from variations in the signal voltage transfer function into the Rx's input impedance, whatever impedance mismatch you have at the Rx end won't matter one jot as far as suppressing the unwanted reverse direction of signal pickup.
Anything coming in from the 'back' will never bounce back to the Rx due to the perfectly matched dummy load termination and any signal received from the forward direction hitting an Rx mismatch will only suffer a portion of its energy being bounced back to the terminated end, never to trouble the Rx again.
[1] After reading that pdf on the mysterious drop in performance of a BOG, I'm guessing that an obvious way to improve performance, as well as reduce variations in Z, would be to lay down a metre wide 100 metre long roll of thin copper sheet along the ground upon which to lay your BOG wire and hammer a pair of grounding rods either side of the copper groundplane, say every ten metres, to bond the copper sheet to ground for good measure.It still won't be perfectly stable with regard to Z but it will come a damn sight closer to the ideal and reduce losses to boot. :-)
A much cheaper, and more practical alternative to this problem might be simply to suspend it a few feet above the ground clutter (say 6 to 10 feet, depending on the nature of the ground clutter and how often you plan on hacking it back).
Incidentally, the Z of a BOG will vary along its length according to variations in the local ground conductivity. This isn't usually a problem since in general, these changes in Z tend to be relatively smooth in nature. It's only step changes in Z that give rise to unwanted reflections that would spoil the immunity to unwanted signals arriving from the back end.
This leads on to the idea of an extremely long BOG not needing to be quite so perfectly matched to its termination dummy load since the increased return losses will improve the situation of unwanted reflections spoiling its rejection of signals originating in the reverse direction.
Given a long enough wire, you can simply bury the last ten yards or so into a shallowly downsloped trench and forget about adding a discrete terminating impedance altogether. Just let the ground do the job for free! :-)