The initial heat transfer rate to the thick copper plates will certainly be greater than the heat transfer rate to the circulating water bath, simply because the conductive heat transfer rate to the copper will be substantially higher than forced convective heat transfer rate to the water. But that does not necessarily mean the hot metal plate will reach some final lower temperature sooner with the thick copper plates than with the circulating bath. It will also depend on the heat capacity of the thick copper plates (a function of their mass) compared to the heat capacity of the circulating water bath (a function of the mass of the water bath).
Not only will the heat transfer rate to the copper plates be higher, but the rate of increase of the temperature of the plates will also be higher because the specific heat (heat capacity per unit mass) of the copper is much lower than the water. Once the copper plates reach the same temperature as the hot metal plate between them, heat transfer stops.
The heat transfer rate to the circulating water bath will be lower, but the rate of increase of the temperature of the water bath will also be lower because of the large heat capacity of water. Only when the temperature of the water bath equals the hot metal, will heat transfer cease.
So the answer to the question posed as the heading of your question (initial rate of cooling) is the copper plates. But if your question is which one, the copper plates or water bath, will reduce the temperature of the hot metal to a specific lower temperature faster, the answer may depend on the total mass of the copper plates vs. the total mass of the water bath, and perhaps some shape factors. It may become a matter of solving two calorimetry problems.
And, of course, all of this will depend on the initial temperatures of the thick copper plates and water bath.
Hope this helps.