you can do that with layers. overlap the layer of the two videos do a fade out and a fade in over the two and end the fade out before it goes to black. as you will be overlapping the exact same background, from what i understand, it will not be an issue.
again just about every video editor ive played with, not many mind you only about 4 or so from final cut to adobe premier, to cinderalla and a few others. if they dont have the built in tool, you can just use layers to get that exact effect.
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