The production of a highly polarized positron beam via nonlinear Breit-Wheeler processes during the interaction of an ultraintense circularly polarized laser pulse with a longitudinally spin-polarized ultrarelativistic electron beam is investigated theoretically. A new Monte Carlo method employing fully spin-resolved quantum probabilities is developed under the local constant field approximation to include three-dimensional polarization effects in strong laser fields. The produced positrons are longitudinally polarized through polarization transferred from the polarized electrons by the medium of high-energy photons. The polarization transfer efficiency can approach 100% for the energetic positrons moving at smaller deflection angles. This method simplifies the postselection procedure to generate high-quality positron beams in further applications. In a feasible scenario, a highly polarized (40%-65%), intense (10^{5}-10^{6}/bunch), collimated (5-70 mrad) positron beam can be obtained in a femtosecond timescale. The longitudinally polarized positron sources are desirable for applications in high-energy physics and material science.