While the duchess might technically have the authority and power to get rid of the maid, she was still in a vulnerable place social wise, so she needed reason/proof to get rid of the maid.
I hope that the scheme was also the authors way of showing how unhinged the maid had become, because if it had been successful, it would have reflected incredibly badly on the duke himself. The most likely assumptions that the guests would have come up with would likely be that either she is innocent and someone was deliberately screwing with her (and thus challenging the duke), or that she was deliberate and making a statement about the duke.
The duke would probably also wonder how the heck she ended up dressed like that and why no servant said a damn thing to her or him.