The production of polarized proton beams with multi-GeV energies in ultraintense laser interaction with targets is studied with three-dimensional particle-in-cell simulations. A near-critical density plasma target with prepolarized proton and tritium ions is considered for the proton acceleration. The prepolarized protons are initially accelerated by laser radiation pressure before injection and further acceleration in a bubblelike wakefield. The temporal dynamics of proton polarization is tracked via the Thomas-Bargmann-Michel-Telegdi equation and it is found that the proton polarization state can be altered by both the laser field and the magnetic component of the wakefield. The dependence of the proton acceleration and polarization on the ratio of the ion species is determined and it is found that the protons can be efficiently accelerated as long as their relative fraction is less than 20%, in which case the bubble size is large enough for the protons to obtain sufficient energy to overcome the bubble injection threshold.