Does the hadith "Indeed, Allah has overlooked for my Ummah what their hearts whisper, so long as they do not act upon it or speak of it" contradict speaking to a psychiatrist about one's intrusive thoughts for the purpose of treatment?

There is no contradiction between seeing a doctor and the Hadith (definition): "Indeed, Allah has pardoned my Ummah for what they whisper to themselves." What the person afflicted with whispers tells the doctor is an account of the whispers that have occurred, and this is forgiven. You should continue with the treatment, seeking Allah's help, relying on Him, and following the doctors' advice.