I use Adobe Auditions, I look for a quiet spot - usually the beginning or end and auditions can take a 'profile' of these background noises - including hiss, once it completes taking the profile you can choose to remove it from the overall tune. For your tune I had to run it twice as there were two 'steps' in the background hiss, an initial quiet hiss followed by a slightly louder hiss (presumably you turned something else on). You're right to be concerned about the highs though, noise removal can have to some really weird effects.
These days, at the end of a song, I carry on recording - just background noise and me, breathing normally - I then take a profile of that.
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