Non-union-affiliated shop steward
Solution 1:
A general term for this is staff representative or staff rep for short. This can be used for any person who represents staff and need not be a member of any union (although they may).
If it really must be stressed the person is not a union member, I'd got for non-union [staff] rep.
Solution 2:
As you say, I don't think there is a well-established word for it. I would go for workers' representative, but it will need a bit of explanation exactly what is meant.