A turbine engine system includes a first lubricant circuit, a second lubricant circuit, a plurality of engine stages and a shaft. The first lubricant circuit includes a first turbine engine component that is fluidly coupled with a first lubricant heat exchanger. The first turbine engine component includes a gear train, which connects a first of the engine stages to a second of the engine stages. The second lubricant circuit includes a second turbine engine component that is fluidly coupled with a second lubricant heat exchanger. The second lubricant circuit is fluidly coupled with the first lubricant circuit, and the second turbine engine component includes a bearing. The shaft is supported by the bearing, and connected to one of the engine stages.