First, Washington State has "Universal Background Checks".
The firearm has to be delivered to a Washington State FFL"to process the sale or transfer as if it is selling or transferring the firearm from its inventory to the purchaser or transferee,..."
HOWEVER ......
(4) This section does not apply to:
(a) A transfer between immediate family members, which for this subsection shall be limited to spouses, domestic partners, parents, parents-in-law, children, siblings, siblings-in-law, grandparents, grandchildren, nieces, nephews, first cousins, aunts, and uncles, that is a bona fide gift or loan;
[ snipped a bunch of other exceptions ]
So, if the firearm is a
GIFT for your brother, then a background check isn't required.
That's the good news,
however (again).....
Since the firearm is a handgun, you are required by federal law to process the transfer through a licensed dealer (FFL) in WA, because this is an interstate transfer. If you are in the military and a legal resident of WA, then you can possibly get around that.
Without knowing your military status and legal state of residence, you should at least qualify for the exemption under the background checks rules. You should be able to go with your brother to an FFL in WA and complete the transfer in one visit. There will be a fee charged, I'm sure.
Washington does allow its residents to buy long guns from out of state and bring them in without going through an FFL. The question then becomes if the seller's state allows it, and does the specific out of state dealer's policy allow it? If they can sell it to you legally, then you can buy it and bring it back to WA legally. Just not for handguns.
I see nothing in the law requiring firearm registration. That's kind of interesting, since it's almost impossible to enforce universal background checks if they don't know who has what firearms. A gun could be passed around (transferred) several times with no background checks, because there's no tracking other than those law-abiding owners who required the buyer to do the background check for private sales.
I do find in the law that felony firearm offenders are required to register with law enforcement, like sex crime offenders here. That actually makes more sense than firearm registration. At least a private seller has the capability to refer to that list to ensure the buyer is not prohibited. Not saying it's a good system, just a better one than ours. My opinion, of course. Put the burden on the criminals, not the "good guys" buying and selling guns.
BTW, the Supreme Court says that criminals and anyone else who is prohibited from owning guns can't be coerced into registering their guns with the state -- it would be self incrimination. So, only people legally allowed to have guns can be charged for not registering one. That doesn't apply to WA since they have no registration, but I thought I'd toss that out since you live in Hawaii.
Anyway, I recommend your brother contact a local WA FFL and verify all this. They are the best source of info since they are effectively enforcing the state's laws on gun transfers. But from looking at their laws, flying with the gun and going through an FFL to process the transfer is the bare minimum you'll have to do. Hopefully the background check will be waived. (Just don't let your brother hand you a check with "Handgun" in the Memo Line, and don't give him a receipt! Repeat after me: "It's a gift."

)
https://app.leg.wa.gov/RCW/default.aspx?cite=9.41https://app.leg.wa.gov/RCW/default.aspx?cite=9.41.113As for flying with it, there are plenty of threads on here about the ever-changing rules for various airlines. The basics are:
1. Use a well-constructed, lockable container. Pelican is always a good choice. Manufacturer's cases are often too flimsy -- they can be pried open on the sides without removing the locks. Put all gun parts in the lockable case, such as mags and grip back straps. Some airlines and states treat any and all firearm parts and ammunition the same as a firearm when flying.
2. Place the unloaded gun in the case and lock it. Use KEYED locks with shackles that are not much longer than needed to close the lock onto the case. Long shackles may allow the case to be partially opened.
3. Do not follow any guides online that say to use a TSA-approved lock. Those locks allow TSA employees to open the case without you present. Keep the keys on you, and insist on being present to unlock the case if inspecting is needed.
4. You'll sign a declaration at the airport stating the firearm is unloaded. Place that inside the gun case if instructed, or place it inside the checked bag the gun case is in.
5. Don't check the case by itself. Place it inside another piece of luggage you are checking. Don't try to take it on the plane as carry-on. Should be obvious, but many people try it all the time. I don't lock the checked bag, since TSA seems to enjoy inspecting my bags containing gun cases several times.
6. Don't put stickers or markings on the gun case or luggage to indicate a gun is inside. That's a good way to get it stolen.
7. Show up at the airport early so you can take care of the declaration process and any inspections without being rushed. Anticipate delays.
8. You may contact the airport for specific instructions, but they normally just refer you to their website which can be out of date. So, don't be surprised when they try to surprise you with changes upon arrival. I'd say call the airport a day or two before departure at least try to get the latest updates. Ask specific questions if they give you confusing or contradictory info compared to the website.
I've only flown United and AA with a firearm, but my daughter used Hawaiian a few years ago. Never had any problems, but as I said, rules keep changing.
P.S. WA has penalties for not safely storing a firearm. They stop short of describing what places and storage conditions are considered "safe". It's more after-the-fact, results-based penalties. As in, if someone steals your gun and commits a crime with it, you could be liable if they decide you didn't take reasonable precautions and stored it safely. That might mean: don't leave the gun in your brother's car while you two stop for a beer.
Wait until after it's been transferred ... THEN go out!
https://app.leg.wa.gov/RCW/default.aspx?cite=9.41.360As usual, take this as a "to the best of my knowledge" answer. You should do your own research: contact the WA police and/or FFL as well as the airline. Take good notes, especially names of who you talk to and when. Then if there's a problem, you can explain where you were misinformed.