Buddhist rAma's name was 'rAma panDita'. LakshmaNa's name was 'lakshmaNa panDita'. 'panDita' in Sanskrit means a scholar. Often persons of priest class are called 'panDitAs'.
No such scholary suffixes. Devotees use hundreds of prefixes such as 'ayOdhyA rAma', 'kOdanDa rAma (in the name of his bow)', 'sAkEta rAma (in the name of his kingdom)', 'kAruNya rAma (compassionate rAma), 'vijaya rAma (victorious rAma), 'sundara rAma (handsome rAma)' etc. 'panDita' is never used, because he belonged to the class of 'kshatriyas (warriors) and not panDitas(scholars)'.
Buddhist rAma didn't have Sita as wife. Apparently he was a bachelor.
vAlmiki rAma married SitA at janakpur, nEpAl, winning her in a bow-wielding contest.
Buddhist rAma panDita didn't kill rAvaNa, because there was no rAvaNa.
vAlmiki rAma killed his wife's abductor 'rAvaNa' and subjected her to a fire-chastity test.
Buddhist rAma didn't have the last brother 'Satrughna'. Total three brothers.
vAlmiki rAma had 'Satrughna' as his youngest brother. He appears as a bride-groom marrying 'SrutakIrti', at the same venue and time as SitA and rAma. His role appears again only in uttara rAmAyaNa. He was sent as the protector of rAma's sacrificial horse.