Cllr Cardownie was only re-elected to his ward in north Edinburgh after eight counts of second and third preference votes from other candidates.
The party chose to run two candidates in Forth and Leith but the tactic diluted support.
Incumbent Rob Munn even lost out to his own running mate and newcomer, Adam McVey, who was elected with 17.6 per cent of the vote.
Speaking after his defeat, Mr Munn said: “There’s always a risk when you stand two candidates and unfortunately it didn’t work out as we’d expected. I’ll now be looking for a job.”