“What’s your edge?” the actor playing Craig Oliver asked Dominic Cummings in Monday night’s TV drama Brexit: The Uncivil War. The programme does a decent job of demonstrating how Vote Leave used a superior slogan (“Take back control”) and false claims (on money for the NHS, and Turkey joining the EU) to capture the public mood. Sadly it brushes over the illegality of their overspending and overplays the role of their data wizards – who are now common in politics.

But rather than looking backwards at 2016, the question posed by Oliver – David Cameron’s then spin doctor – is perhaps more pertinent with regard to the prospect of a new referendum. Some bookmakers have slashed the odds of a people’s vote to virtually evens as Theresa May’s deal looks destined to fail.

Although my part in the TV drama was blink-and-you-miss-it, in reality I was part of the remain campaign from the start. My job was to build an organisation – Britain Stronger in Europe – from scratch, which ultimately employed 200 people around the country. I was there in 2015 when Oliver and the rest of Cameron’s team wanted nothing to do with us as they pursued their hopeless renegotiation of EU membership. And I was there when, despite cordial relations, the No 10 team focused the campaign increasingly narrowly on economic risk.

So what are the lessons if another referendum becomes the only route out of the current mess?

Gordon Brown’s video from the ruins of Coventry Cathedral was the most shared remain video during the referendum

First, a new campaign will need an emotionally resonant message rather than relying solely on the “facts”. The former prime minister Gordon Brown came closest to finding this tone when he argued that Britain should “lead not leave”. His video from the ruins of Coventry Cathedral making this case was the most shared remain video on Facebook during the referendum. Unfortunately, greater use of this slogan was dismissed by our pollsters. They insisted that voters simply wanted facts on the economic impact. “Tell them again” is likely to be the leave slogan if there is a new vote. Something appropriately patriotic and uplifting will be needed on the remain side to compete.

Second, political leadership must come from all major parties. Labour’s refusal to join the official campaign meant that remain was dominated by David Cameron and George Osborne. Labour’s logic was that sharing a platform with the Tories in the 2014 Scottish referendum had been fatal for the party, which was all but wiped out in Scotland at the next election. The consequences were that neither Jeremy Corbyn, John McDonnell nor even Alan Johnson, who led the independent “Labour In” campaign, were part of remain’s central command. All the key decisions were taken by a small group close to the then-prime minister, and relayed back to campaign staff. Indeed, Peter Mandelson denies that the call with David Cameron portrayed in the TV drama ever took place.

The No 10 team were full of hubris from their surprise (but narrow) general election victory, and thought they had all the answers. Having Labour at the top table would have forced a more balanced campaign and prevented groupthink.

Third, the campaign must be prepared to take on its opponents’ arguments. Immigration was the dominant theme in the final month, but No 10 wanted to focus purely on the economy. Instead, we should have made the case for free movement’s role in supporting the NHS and providing opportunities for Britons, especially young people, to travel, study and work abroad.

Fourth, it doesn’t take a brainbox in a broom cupboard to know that the campaign will need the ability to frame its opponents. Leave did a superb job of painting remain as the establishment “Project Fear” campaign. This was patently absurd given the scare stories on Turkey and the backgrounds of Boris Johnson, Jacob Rees-Mogg and even Cummings (the Vote Leave campaign director), who married the daughter of a baronet. But it stuck.

Remain’s creative agencies produced a series of eye-catching adverts that would have nailed leading leavers in the public’s imagination. This included one of Boris Johnson in the pocket of Nigel Farage, mimicking the successful poster of Ed Miliband and Alex Salmond the year before. Sadly, No 10 banned the images because, as Oliver’s character reveals in the drama, they had one eye on putting the Tory party back together following the vote, rather than both eyes on the prize.

Finally, it is critical to use every available resource to make your case to voters. The leading leave and remain campaigns were each able to spend £7m. We know now that Vote Leave broke electoral law by channelling an extra £625,000 to one of its offshoots. But leave also managed to spend twice as much as remain on digital adverts. It did so because remain wasted money on unnecessary vanity items like the daily tracker poll that Cameron wanted, and on inflated fees for American consultants. Discipline is critical in any campaign, and remain was bloated.

After watching Monday night’s drama, a new referendum may be the last thing that many people want. But if it does happen, it must be leave and not remain asking who has the edge.

• Will Straw was executive director of Britain Stronger in Europe, and is on the board for Open Britain