It looks like the remote switches are normally open. So your addition of a relay would not effect the on machine operation.
However I would:
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Use a simple toggle switch on your device to select forward / reverse.
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I would not use one of the arduino relays.....much too noisy. I suggest you purchase a small reed switch relay. Perhaps one of these
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I would add a button on your device for one at a time (aka manual)
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I would and another switch for timed, manual
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I would add a device (pot or encoder ) for time adjustment.
John