The Green party is pressing broadcasters to ensure its leader, Natalie Bennett, is included in any TV election debate alongside David Cameron, Ed Miliband and others .

The move comes amid a surge of new popular support, particularly in Scotland, where a previously tiny membership has more than trebled since last week’s vote on independence. Being at the top TV table is seen as a way of cementing the electoral status it feels it deserves. It would also give the party a major platform to tap prospective votes from legions of people alienated from Westminster politics.

“We think we should be there. We think that it would be absolutely transformative,” Bennett said on Wednesday. “There’s a significant chance the debates won’t happen at all. But if they do, the Tories are going to push hard for us to be on at least one of them. Cameron wants us to be there as well as Ukip. Cameron is basically not going to let [Nigel] Farage on without me.”

The Greens have complained that they has been polling the same level of support as the Lib Dems, 7%, and yet struggle to obtain extensive media coverage beyond anti-fracking or climate change protests.

That only reinforces the idea that this is a single-issue party with little interest beyond the environment, unlike some counterparts in Europe, notably the German Greens.

In particular, Bennett and the party’s lone MP but best-known figure, Caroline Lucas, are keen to further increase membership, which has already surged by 40% since 1 January and is expected to hit 20,000 in England and Wales in the next month. Membership in Scotland, where the party is organised independently, has increased by 4,000 to 5,600 over the past week alone. Interest in the Scottish referendum has given the Greens renewed hope about voters’ desire for wider and widespread political change.

Bennett believes party policies to introduce an annual wealth tax on people with assets of more than £3m, a £10 minimum wage by 2020, rail nationalisation alongside opposition to free schools and academies will strike a chord with the public.

On schools, Bennett said: “We want a different sort of education system that educates children and does not think of them as future workers. We start education too soon.

“We call for formal education to start at least one year later. It is to get away from this idea that we have a sausage machine that we are just shoving children through and turning them into this identikit person.”

She added: “We have to get away from the idea that schools can somehow make up for the incredible levels of child poverty. We have one of the most unequal societies and some of the highest levels of child poverty. The test results just reflect that. Saying to schools that you have got to solve this problem is just unrealistic.”

Bennett is happy to position the party to the left of Labour, saying: “I think there’s a huge political space that’s vacant apart from us”, but is less keen to charge into quick changes on constitutional reform. “English votes for English laws would result in absolute chaos,” she said, adding that the Greens wanted a people’s constitutional convention to consider a range of issues including the future of the House of Lords.

North of the border the party believes it is on a roll and wants to ensure that promises of devolving power from Westminister result in a different and better country. Alison Johnstone, a Green member of the Scottish parliament for Lothian said: “Scotland has become a participative democracy. I feel encouraged and optimistic.”