According EASA you don't need a FD for RNAV 1 and 5, however, according the Boeing RNP capability document for B737 you do need a FD for all PBN specifications except RNAV 5.
My company interprets this as FD with LNAV function (and VNAV for RNP APCH with VNAV minumums and RNP AR APCH). Thus, with LNAV not available we would be limited to RNAV 5 and conventional approach.