The CBI is facing a mounting crisis in its membership, with the Law Society of Scotland (LSS) and three other public bodies quitting on Tuesday to protect their neutrality in the Scottish referendum debate.

The LSS became the latest high-profile organisation to resign after the CBI registered with the Electoral Commission for the no campaign in the independence referendum. Public bodies have said rules preclude their membership of a political organisation, and some businesses have quit in opposition to the anti-independence stance.

The LSS was quickly followed by two universities, Strathclyde and Glasgow Caledonian, and the quango Skills Development Scotland.

The CBI had 80 members headquartered in Scotland. Eight, including VisitScotland and Edinburgh University, quit over the bank holiday weekend, and another two are expected to follow on Wednesday.

The CBI insists it was legally obliged to register as it had already publicly declared that it opposed independence and planned to make that clear in the runup to September's referendum.

John Cridland, its director general, said it was a compliance issue. "We have to operate within the law," he told BBC Radio 4's Today programme on Monday. "The decision we took was that simply to do our normal activities on behalf of our members, including events and public statements between now and the referendum, we were advised that we needed to comply with the Electoral Commission's rules because we have a position on the issues."

Lorna Jack, chief executive of the LSS, which represents Scotland's solicitors, said: "We do not believe we could credibly retain our impartiality whilst being a member of and actively contributing to another organisation which is formally registered with the Electoral Commission to campaign for a no vote. That is why we have resigned from the CBI today."

Registering with the Electoral Commission allows organisations to spend up to £150,000 during the 16-week campaign. The CBI is one of the first to register, along with the official campaigns Yes Scotland and Better Together, the Tory party and a small anti-independence group in the Borders.

Dozens of other pro-independence and pro-UK groups, likely to include many trade unions expected to support a no vote, will be expected to register before 30 May, when the formal campaign period starts.

The CBI refused to comment on the latest resignations. On Monday three of Scotland's largest universities – Edinburgh, Aberdeen and Glasgow – quit on the same grounds.

Executives at Dundee and Heriot Watt universities are expected to follow, but Robert Gordon University in Aberdeen has said it will stay with the CBI because it wants to keep in close contact with business and industry.

The broadcaster STV resigned over the weekend, saying its neutrality could be compromised. Two companies whose directors have direct links to the pro-independence movement, Balhousie Care and Aquamarine Power, also resigned, accusing the CBI of failing to fully consult its Scottish members.

Strathclyde University's decision on Tuesday was doubly embarrassing as the CBI Scotland director Iain McMillan is chairman of Strathclyde business school's advisory board.