BobL, the argument about using 'man' or 'men' as an inclusive term has been, since the 1960s, that it isn't and never has been neutral, and goes back to a culture where the man was the head of the family and all members of that family shared his extended identity. For example (and I can't remember now which book I read this in back in the day), there are 50 people in a room, 49 women and 1 man. We use the sentence, 'There are 50 men in a room', thinking it is neutral. The women have disappeared.

LadyThorn, you have two great difficulties. The problem with a song like One Man's Hands is that it is so familiar in its original form that changing the words is likely to jar severely. Making something that sounds good enough fit the scansion seems impossible: Each one's hands? A person's hands? etc. etc. They all sound awful. There are plenty of good songs to be sung at rallies. If it were me, I'd explain and choose another.