Haim do a fine line in what you can only call “Haim karaoke”. Night three of the UK leg of the LA band’s Sister Sister Sister tour finds the lead singer and guitarist Danielle Haim suffering from flu. We know this because her younger sister, Alana Haim, is telling us – while “hiding” behind the silent Danielle, and using mime hand gestures to make it look like Danielle is talking. A doctor has prescribed Danielle strict bed rest. (“Awwww,” says the crowd.)

Alana’s hand, however, flips a V at the notion of bed rest, and at cancelling Haim’s two sold-out Dublin gigs. The LA three-piece (with two more in the live line-up) play on tonight – but without the vocals of their lead singer, who, despite a sleek purple suit, looks at her least Californian tonight: grey-pale. You imagine the band are hoping to conserve Danielle’s vocal cords for the two more sold-out nights they are to play at London’s Alexandra Palace at the weekend, where a total of 20,000 tickets were snapped up quickly, despite the middling sales performance of the band’s second album, Something to Tell You.

There’s an odd disconnect too, tonight, between speaking out and shutting up: Danielle is finding herself muted on the day the news hits that the band sacked their former booking agent when they discovered that Haim had been paid 10 times less for a festival performance than a male act just one line up the billing.

So this set’s vocals are divided up between Alana (youngest sister, keyboard and guitar) and Este (eldest sister, on bass and enduringly hokey facial expressions). Back in February, Este was briefly rechristened “mystery drunk woman” at the Brit awards, after she made faces when Liam Payne, a former boy band member, and Cheryl, a singer who has been often married, were being interviewed. (“Not drunk, just living my truth,” Este later tweeted.)

Este kicks off the whole Haim-sing-the-songs-of-Haim scenario with Falling, from the band’s widely adored 2013 debut album, Days Are Gone, while Alana takes the reins on Don’t Save Me, another debut album throwback. Back and forth they go, through a set that flies by in a blur of word-perfect crowd singalongs and onstage repartee. There is a show-must-go-on feel here, born from the sisters’ past as a family covers band. And there is an argument that Haim’s harmonies are so ever-present, it doesn’t make all that much difference who is singing lead.

You do miss Danielle, though, who confines herself to eloquent guitar solos, the odd bit of drum-bashing and mouthing the words enthusiastically. All the Haim sisters are multi-instrumentalists who sing, and most Haim songs have multiple vocal parts, the better to meld the harmonies of 90s R&B and 70s west coast rock – their special superpower. The tonal absence of Danielle is disorienting, rather than disastrous, but there is a marked difference: the middle Haim packs a gravitas her two sisters lack.

Alana seizes the spotlight tonight. Demanding Guinness – not a tactic likely to alienate a Dublin crowd – she tells the tale of how she advocated for the song that eventually became the first single from Something to Tell You. Her sisters didn’t like it, though, until Alana – dialling up the youngest sibling angst – came up with a chord that transformed the song. Este then “sprinkles some truth” on “this bullshit sundae” – apparently, all the song needed was slap bass and a wise head. It all builds up very slickly to Want You Back, a song packed with emotional IQ and vocal interplay; Alana handles the lead.

If we’re going granular, Este “does” Danielle marginally more convincingly than Alana. Her range runs deeper, with an added measure of huskiness. Her closing take on Right Now is, perhaps, the song of the night, with added extemporisations about exes who blow hot and cold.

But Right Now and its equally lush album-mates only point up the conundrum why Something to Tell You didn’t quite hit the ubiquity of Days Are Gone. Second albums often suffer after shock of the new is gone. Perhaps the gap between the two records – four years – was a major factor in these gadfly times. It was later revealed that one major reason Haim’s second was delayed was that Danielle’s other half – super-producer Ariel Rechtshaid – had been diagnosed with testicular cancer; no one was in the mood to produce breezy AOR about the minutiae of relationships for some time.

Rechtshaid, though, has recovered. And now Haim seem poised to take on their rightful place as the summer festival band in excelsis, with – hopefully – their lead singer back front and centre.