Did they even search for a an actor who had a similar background to portray the role or did it go to Stone because she was "hot" at the moment?
I would guess they did because on the first part of "The Making Of Aloha" DVD extra, they talk a fair bit about who would play Bumpy. They searched and searched and couldn't seem to find someone right to do the role. Finally, Crowe went to Bumpy himself and said, "Is there any chance...?" And Bumpy said yes. They show Bumpy and Bradley filming a scene and when it becomes too difficult for Bumpy to continue, he breaks character and Bradley is so supportive and tells him what a great job he is doing. But for the role of Allison, I'm guessing you are correct in that for the two lead roles, they wanted two people who could draw in an audience.
Also, Crowe has said that he began writing this story in 2007, so there was seven years for them to consider actresses who were one quarter Hawaiian, one quarter Chinese and half white. Since they worked so hard to fill the much smaller role of Bumpy, I would assume they put in as much effort to find the best possible choice for Allison Ng.
The film's Wikipedia entry says she was the first to be cast so my guess is no and that's where the problem lies. A white actor shouldn't have been cast in that role period in my opinion.
The only Hawaiian actress I'm aware of is Tia Carrere. Tia was born in Hawaii and is of Filipino and Chinese descent. But she's 22 years older than Emma Stone and the character of Allison has her whole career ahead of her, so the age was wrong. Plus Carrere didn't have the box office clout of Stone, whose casting helped Crowe get the 37 to 52 million budget (as stated at wikipedia) that he needed to get this movie made.
As someone who is mixed but of Native American/White ancestry I can tell you there are a lot of us who feel that we don't "fit" into either culture we belong too and I'm sure with a casting search they could have found a more appropriate choice. But with that choice they would have lost the buzz that came with casting Stone.
As a member of the LGBTQ+ community, I wonder about straight people playing LGBTQ+ roles. I don't remember anyone voicing objections to the casting of these people/roles:
Brendan Fraser in The Whale
Timothée Chalamet (and Armie Hammer) in Call Me By Your Name
Cate Blanchett in Carol
Taron Egerton in Rocketman
Saoirse Ronan and Kate Winslet in Ammonite
Nick Robinson in Love, Simon
Ewan McGregor in Halston
Eddie Redmayne in The Danish Girl
Colin Firth in A Single Man (and Supernova)
Charlize Theron (and Christina Ricci) in Monster
Jake Gyllenhaal and Heath Ledger in Brokeback Mountain
Rami Malek in Bohemian Rhapsody
Annette Bening and Julianne Moore in The Kids Are All Right
Sean Penn in Milk
and
Stanley Tucci in The Devil Wears Prada
Stanley Tucci in Burlesque
Stanley Tucci in Supernova
Stanley Tucci in The Daytrippers
Stanley Tucci in Little Chaos
And these are only the bigger names in movies of recent memory. If we go back to the 90's and earlier...

The list will get much longer!!