The remaining 28 councils are to pay an extra 3% from the beginning of April while a 'root and branch' review is undertaken. The review, expected to include redundancies, will be completed at the end of June.
The organisation is unable to continue in its current form following the withdrawal of City of Glasgow Council, Clackmannanshire Council, Falkirk Council and Dundee City Council.
The withdrawals means the loss of£470,000 from the convention's budget of around£2.5m.
It is hoped the latest deal will secure the short-term membership of those councils reviewing their fees.
Convention president Norman Murray said: 'The review will be undertaken by a wide-ranging group of people who will be talking to member councils about their views of the organisation as it exists.'
Clackmannanshire leader Keith Brown (SNP) said he is taking information from the recent leaders meeting to the full council for consideration.
He added: 'I couldn't support the motion because my council has already decided to come out.'
Falkirk leader John Connolly (Lab) said he was expecting a letter from the president asking the council to reconsider its position.
Falkirk would not be able to alter its decision within six months without amending a standing order.
Mr Connolly said: 'The letter has not arrived and we need it as soon as possible if the council is to reconsider. COSLA should not alter its current budget, but an extensive review is desperately needed.'
Dundee City Council leader Julie Sturrock (Lab) said the issue is highly political. Labour holds 13 of 29 seats at Dundee while the Scottish National Party holds 11 seats.
SNP controlled Angus Council is 'hanging on by the skin of its teeth' according to leader Rob Murray.
The party's shadow local government minister Kenny Gibson said Angus could leave at any time.
Mr Gibson was due to meet Norman Murray yesterday to discuss the organisation's future.