I presume that several members of the Council know one or both of the candidates, as this is the norm. You're voting to co-opt them onto the Council, so you have a pecuniary interest only if their co-option would benefit you or a member of your immediate family financially. If not, you have no pecuniary interest.
If you are an associate of one or both of the candidates, you could declare a non-pecuniary interest, although I can't see why this would be relevant to the matter in hand. Your code of conduct will dictate how non-pecs are handled, but it is unlikely that you will be prevented from participating in the discussion or vote.
Why were the candidates and public excluded from the meeting? The grounds for exclusion are clearly set out in the legislation and the resolution to exclude must state which of the reasons is being applied. This must also appear in the minutes.
As for breaking the Council, perhaps you could suggest that the Chair reads the section of the standing orders relating to dispensations.