A one-man musical about one of the least sympathetic campaign groups of our time, Fathers 4 Justice? The heart doesn't soar at the thought, frankly. But while this new musical of one man's fight to see his child has its flaws, it's also touchingly human, musically sprightly, and performed with appealing agreeableness by Michael Rouse.

Colin Bradley is in court, asking for custody of his treasured little girl Emily, rather than her moving with her mum to LA. In Rouse's sympathetic but self-deprecating performance, Colin is a dead ordinary bloke driven to desperate measures by love. He climbed Big Ben dressed as Robin (bathetically, they'd run out of Batman suits) a few weeks before his trial, in order to – well, what? Raise awareness of the tough time fathers get in the courts? The show certainly presents a one-sided view of custody battles. But it's also a matter of proving himself – although the uninterrogated notion that Colin is more of man, more of a father, for finally getting out there and "fighting" for something is a bit basic.

But Rouse is really heartbreaking at conveying the extent of Colin's love for his child, as he explains what happened to lead him to this situation. Under Adam Lenson's direction, Superhero is frequently tender and moving in expressing this love – and you realise how rarely you hear about father-daughter relationships on stage, let alone in song. While Joseph Finlay and Richy Hughes' music and lyrics include a fair few heartsore or rousingly impassioned numbers, they largely avoid schmaltzy sentimentality, Colin undercutting himself before it gets too high-blown. With a band of just three on piano, bass and percussion, the score is often a pleasingly energetic, inventive thing; staccato bursts of xylophone used to evoke a fearful racing mind, or standing in for the voice of his daughter.

The balance between the songs and the book by Michael Conley feels a bit off, however. An engaging tale delivered by an engaging performer, I wanted more: too often it segues into the generalised emotion of a musical number rather than getting into the nitty gritty of character. Chrissie, the wife, remains a blank – we neither really get the good times of their 16-year marriage nor understand how exactly they went sour. Superhero could use a lot more detail; while Rouse invests Colin with plenty of nuanced humanity, you're aware you're only getting one side of a painful story.

Still, its portrayal of devoted fatherhood – wanting to do the right thing, terrified of getting it wrong – seems to strike a chord. I'm sure many parents will identify with Colin's self-flagellating and often amusingly shambolic attempts to be the perfect dad, always falling short. And also with his moving conclusion that it's not just kids who need dads – it's dads who need their kids.

Superhero runs at Southwark Playhouse until 22 July.