Lauren Mayberry’s hamster is dead. The Chvrches singer is grateful for the pictures she gets, from fans, of hamsters that look similar to the recently departed Gilbert, and for the messages asking how he is. She was pleased with the stuffed hamster dressed as a dragon that a fan brought her in Japan. But she doesn’t quite know how to break the big news. “People bring him up and I don’t want to be like, [whispers] ‘He’s dead.’ But he is now deceased. He had a good innings. He had a long, fruitful life.” Her bandmate Martin Doherty, known as Dok, adds his own tribute. “That fucker lived like a king. King Gilbert I.” The group’s third member, Iain Cook, nods gravely.

The electro-pop trio are sitting on a hill in a Glasgow park known locally as “the beach”. Or “the jakey”, depending on who you ask. Doherty explains that it’s where local drinkers spend their afternoons. “Don’t be alarmed,” he says. “Just so you know what you’re getting yourself into.” But it turns out what we’re getting into is a picture-postcard version of Scotland: there are no drinkers, just teenage bagpipers, all over the park, playing their own individual tunes. Not that people see Chvrches as a Scottish band, necessarily. “People bring it up in America, where a lot of people have Scottish roots,” Mayberry says. “But in the UK, not so much.” “I certainly felt like a Scottish band when we didn’t announce a Glasgow show the other day,” Doherty smiles. (They say there will be one soon.)

Time in their hometown is increasingly rare for Chvrches. They’ve spent the past three years slowly and steadily building towards remarkable success. Their debut album, The Bones of What You Believe, which came out in 2013, is edging towards a million copies sold (“But if we bust a million, I’ve got to get a tattoo, so I don’t want to do that,” Doherty says). The irrepressible single The Mother We Share was everywhere in the UK and went gold in the US, where the band are touring relentlessly, packing out increasingly large venues. They are so big in Japan that fans bring them gifts of toy hamsters. They’re officially a massive pop band. But what’s more remarkable is that they have done it in a relatively old-fashioned way: by working hard, sticking to their guns and refusing to take compromising shortcuts.

“We could have sold 200,000 more albums if we’d hidden Iain and I from view and put Lauren on the cover of every magazine. But there’s 100 of those acts and that stuff goes away,” Doherty says, firmly. “We ended up doing it in an indie band style. We broke through via word of mouth. It was about doing it in an honest, right way.”

That way involved learning the power of saying “no”: to the photoshoots that wanted to push the men to the back, to co-writers, to outside producers, to the women’s magazines who wanted to interview Mayberry on her own. “We were making music that was slightly poppier than any of us had done previously,” she explains. “I didn’t want us to fit into a cookie-cutter mould of what that kind of band is meant to be like. I don’t want to be the front for somebody else’s creativity and sell that day in, day out.”

But now, as they prepare to release their second album, Every Open Eye, they are relaxing the rules. Last month, Mayberry was interviewed by New York magazine alone, away from the two men. Their latest video, for Leave a Trace, is a glossy, expensive-looking beast, largely focusing on the singer, with Doherty and Cook looking, well, slightly blurry.

“We wanted to establish [Chvrches] as a band first and foremost, and have that base,” Mayberry explains. “Then you can move on from that. We’ve done a couple of women’s mags but we tend to talk about feminism and women in the industry, which I feel more comfortable talking about. It’s a more valuable discussion than, ‘Oh, you’re a girl in a band, what hair conditioner do you use?’ I use hair conditioner, and I like talking about it. But I don’t want that to be the question.”

Surprisingly, one of the catalysts for Mayberry’s determination to do it the right way is a bruising early adoration for a certain Canadian pop-punk. “When I was growing up, there were things that were sold to you in a certain way …” she begins.

“Are we talking about Avril Lavigne?” Doherty asks, in a tone that suggests he has heard this before.

“I was totally the target audience when that came out,” she continues. “The marketing campaign was: ‘She’s just like you. She writes the songs.’ To a teenage girl writing songs in her bedroom, that’s amazing. And then I woke up one day and realised it was all written by the Matrix [songwriting and production team]. Teenage me still loves that record [Let Go], but why did they sell it like that? I was pissed off that it was being sold as real.”

It’s easy to imagine the teenage Mayberry being outraged by the Lavigne deception; she’s driven by an obvious fighting spirit. There can’t be many artists on the Billboard charts, for example, who founded a feminist collective called TYCI (it stands for Tuck Your Cunt In, and she still edits many of the pieces that run on its site and in its zine). In 2013, she wrote a piece for the Guardian, exasperated by the amount of sexually violent threats thrown her way on social media. “Why should I feel violated, uncomfortable and demeaned? Why should we all keep quiet?” it read.

“It was weird to see how it spread,” she says now. “It was reposted on most music websites and news outlets. It was mad.” Did it make a difference to the level of abuse she received? “Erm ...” She pauses. “I guess we figured out how to switch off the DM function. The community self-polices a bit more. I operate a pretty strict muting and blocking policy on Twitter. If I get one that’s especially aggressive and I don’t feel emotionally equipped to deal with it, then I don’t get upset about it. I think: ‘What would Lauren from Chvrches do, as opposed to me?’ Then I think: ‘Cool, fuck you,’ block that person, carry on.” She likens this ruthless, efficient alter ego to a superhero. “My band persona is 25% tougher than I am. In that moment, I can step it up.” Depressingly, in the days after we talk, she proves the abuse is still coming when she tweets a link to a disgusting 4chan thread about their latest video. “Dear anyone who thinks misogyny isn’t real. It is and this is what it looks like.”

The band took some much needed time out at the start of the year, recording their second album in the Glasgow studio they’ve always worked in, which used to be Cook’s, and now belongs to all of them. “Same shite studio, with slightly nicer paint and more synthesisers,” Doherty says. “It’s a nice studio now,” Cook protests. (They’ve sold a lot of records in the US, after all.) “I would describe it as decent now,” says Doc. “Before, it was kind of a shite studio. No offence!” They came up with Every Open Eye quickly; so quickly that it took them by surprise. “There’s always a danger of overthinking a second record but it wasn’t something we discussed either stylistically or thematically beforehand,” says Cook. “We wanted it to sound and to feel spontaneous. We wanted to go with our noses, the way we did on the very first day we started working together.”

It sounds like a natural successor to their debut: still recognisably Chvrches, mixing sweet pop synths with deceptively barbed lyrics, but bigger in every way – it feels built to fill the increasingly sizeable venues the band have found themselves headlining. “I’m not bullshitting,” Doherty says, “we weren’t out to write those songs. You can’t come in and go, we absolutely need a We Found Love. We’ve tried that – not in a cynical way, just as an experiment.” He shrugs. It’s not for them. “Whenever anyone says let’s write a banger or let’s write a hit, you can absolutely guarantee that it’s the opposite of what you’re going to get.”

But, I say, the album is full of bangers. “Accidental bangers!” says Mayberry. “That’s album No 3,” Cook laughs.

• This article was corrected on 20 August 2015. An earlier version referred to Chvrches’ album title Every Open Eye as Every Eye Open.

