This can be handled using Arc Hydro "deranged" terrain processing workflow where the sinks are known. In your case, known sinks are the lake polygons (we use term "deranged polygon" for those). There are few Arc Hydro documents that explain the processing workflow when dealing with deranged terrains. Check out "Arc Hydro: Identifying and Managing Sinks" and "Arc Hydro: Overview of Terrain Preprocessing Workflows". Since you already know where your polygons of interest are, the workflow is captured in "Use case 3" in the "Arc Hydro: Overview of Terrain Preprocessing Workflows" document.
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