The main campaign group seeking to take Britain out of the EU is in danger of losing the referendum unless “damaging and unnecessary” bickering is stopped, according to a leaked internal email.

In a sign of the bitter infighting in the Vote Leave group, one of its main supporters has accused campaign director Dominic Cummings of undermining the organisation by generating “ill feeling” among workers.

John Mills, the multi-millionaire Labour donor who is the group’s deputy chairman, said that the feuding has prompted the MP Kate Hoey to stand down as co-chair of the Labour Leave group. Hoey, who is now supporting the separate Grassroots Out (GO), has agreed not to say anything publicly about the split.

Mills turned on Cummings in an email sent on Tuesday. Mills was infuriated after Cummings reportedly sent a text which allegedly made disparaging remarks about the former Labour minister and Gordon Brown supporter Nigel Griffiths, who is a leading figure in Labour Leave.

In an email to Cummings and the Vote Leave chief executive, Matthew Elliott, Mills wrote: “What on earth are you doing, generating more and more ill feeling like this entirely unnecessarily? I thought you had promised to stop doing this sort of thing. Don’t you realise that this kind of behaviour puts more and more damaging and unnecessary strain on everyone?”

Cummings, a cerebral figure who helped to create the free schools programme as senior special adviser to Michael Gove during Gove’s time as education secretary, has upset Tory MPs, who complain that he has an abrasive manner. He is refusing to countenance any cooperation with the other main anti-EU group, Leave.EU, which is funded by the Ukip donor Arron Banks.

The feuding within Vote Leave is significant because it could jeopardise its attempt to win official designation by the Electoral Commission as the main anti-EU campaign group. The Electoral Commission will designate one campaign group on either side of the referendum, once the government sets the date of the referendum in secondary legislation.

Mills voices fears that the feuding could boost the pro-EU camp by leading the Electoral Commission to designate Leave.EU as the main vote-leave group.

Polling suggests that a campaign involving Nigel Farage, the Ukip leader, would fail to win wavering voters in the middle ground. Mills is keen to improve cooperation with Leave.EU, whose founder, Banks, is also funding the GO group supported by Hoey. But Mills believes that a referendum win could be jeopardised if Leave.EU is designated as the official campaign.

The Labour donor wrote to Cummings and Elliott: “The bottom line is that Labour Leave are fed up with the way they have been treated by VL [Vote Leave] and the – intransigent and insensitive, from their perspective – policies it pursues –unfortunately typified by the remarks referred to above.

“I did my best to explain why VL has the policies it has and to stress that VL had to win the designation if we were to have any hope of winning the referendum, and this meant that LL [Labour Leave] had to go on being in the VL camp.”

The fightback by Cummings, who has agreed with Elliott to stand down from the Vote Leave board while both retaining their roles, has fuelled speculation that he might be able to deliver his former boss Gove to front the campaign.

Allies of Gove told the Times that he is wrestling with his conscience over which way to jump in the referendum campaign. Gove is highly Eurosceptic, though he will know that losing the referendum would probably terminate the political careers of his friends David Cameron and George Osborne, who want to keep the UK in a reformed EU.

Downing Street has been confident that Gove will support the prime minister and the chancellor in the remain campaign. But rumours that he will support Vote Leave, once collective cabinet responsibility is lifted, if the prime minister secures a deal with EU leaders, were fuelled by the Times piece.

Rupert Murdoch, who is close to Gove, tweeted: “Cameron’s deal with EU a nothing. How can sensible cabinet colleagues accept this? Loyalty to country more important than friendships.”

Mills stood down as the Vote Leave chairman this week, making way for the former chancellor Lord Lawson of Blaby. He is now the group’s deputy chairman. A Vote Leave spokesman said of his leaked email: “Vote Leave is a genuinely cross-party campaign and we value the support we get from our Labour MPs and members across the country.”