If management insists on it, I would have them put it in writing and if something happened, they would have a real problem on their hands because they have assumed the risk of risking losing money by just having one employee responsible. Personally, I would not want to be the person who had that responsibility solely on my own. Even with limited staff the whole purpose of dual control is so that if there is a problem, two people have had eyes on it and personally when I was one of the dual control people, I never took my eyes off the money or let the other person leave me by myself.
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Define Success on your own terms, achieve it by your own rules, and build a life you are proud of. Anne Sweeney