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Trigonal PtBi$_2$ is one of the attractive van-der-Waals materials because of the enhancement of its superconducting transition temperature $T_{\rm{c}}$ by doping chalcogen elements such as Se and Te. Recently, it has been reported that $T_{\rm{c}}$ of Pt(Bi$_{1-x}$Se$_x$)$_2$ is enhanced by a factor of 4, compared to the pristine PtBi$_2$, together with the polar-nonpolar structural phase transition. Thus, it is desirable to study electrical transport properties for this new superconducting compound. Here, we have performed magnetotransport measurements for Pt(Bi$_{1-x}$Se$_x$)$_2$ ($x$ = 0.06 and 0.08) thin-film devices and have observed a peculiar magnetoresistance where a finite hysteresis appears when the superconducting state is broken. By measuring the magnetoresistance systematically, we have attributed this magnetoresistance to the two-level critical-state where fluxons pinned in Pt(Bi$_{1-x}$Se$_x$)$_2$ play an important role., Comment: 7 pages, 5 figures