The most obvious opportunity for profit would be for the NGO to accept payments from non-abused parties to fulfill the murder.
So let's say I want to kill my boss, and my boss is an abuser who the NGO has the murder-right on. I'm willing to pay the NGO 1 million dollars to fulfill that murder.
But as soon as this enters the equation, the abuser is being murdered for non-abuse reasons, since only the murder right is transferable, not the murder reason. A separate case could (and should) be filed against me for conspiracy to commit murder, and I wouldn't be able to claim domestic abuse.
The only assumption we are making in this scenario is that an abused's right to murder their abuser is transferable. Just because this very dark assumption is true in this scenario does not imply that any other dark assumptions can be made. We assume the NGO must be able to profit from this murder in a legal way. I will accept the assumption that being able to sell your murder-right means that the murder-right can be legally purchased, and thus the murder itself would be legal, but I would limit that assumption to the murdering of the abuser in a pre-determined "legal" way, such as the murder being done in a way that would be deemed ethically and morally sound.
A starting off point for what "legal ethical" murder would look is the Geneva convention and other global military agreements on warfare. This means the abuser can't be tortured, can't be treated in a cruel or unusual way, can't be killed via biochemical agents (like sarin or anthrax, etc).
I would say that live-streaming isn't an option because for one it would violate the abuser's right to privacy (which they would maintain before and after death, like anyone else), and it would probably violate laws regarding creating a violent scenario for the purpose of profit, something along the lines of what makes Bum Fights potentially illegal. If we assume a world where public execution is reintroduced, I would argue that it has been reintroduced as a right for anyone to attend an execution, making it some sort of public service, and thus not something you could make prohibitive to attend by charging money. If we think of it like airwaves as being a public good, at best someone might profit from filming the event and providing access to that stream, but that wouldn't be the NGO (directly) but more like YouTube or ABC (it would be a media market, not a direct "we kill em, you pay us to watch us kill em" market).
As for making it an auction on who gets to kill the abuser, I would argue that if the murder-right is transferable but expected to be done ethically, that those committing said murder would likely need some form of licensing, demonstrating they are capable and qualified, etc. We are letting these people commit legal murder, so they need to be registered in some way to do this. This would also rule out the "human safari" idea, as it should. If we introduce some sort of Running Man scenario where anyone who sees the abuser can kill them, we open the door to a lot of liability. We have no assurance that the murderer will murder the abuser in an effective way (what if they just maim them, and the abuser runs off, does that person get away with stabbing someone in the leg just because the person was an abuser? And how can we know for sure that people wouldn't pay just to "try to kill" the person, with no good faith that they will actually do so?)
So if the person is licensed to murder, we're looking at a sort of Uber for murdering. The NGO is the company that buys the rights to murder, the murderer is a free-agent licensed by the state to facilitate said murder. Which sounds like something people would register for to make money, not to pay for the privilege of murdering someone.
Harvesting organs is a clever idea, but would still require a law that made it legal to do so. Otherwise the abuser still has the rights to their organs and what is done with them when they die.
The most obvious opportunity for profit, with all of the above taken into consideration, would be via a life insurance policy. Something similar to companies that exist now that buy life insurance policies from people (especially the sick or elderly) and pay them a lump sum percentage of the policy now and change the company to the benefactor.
This wouldn't count as insurance fraud (though it may drive up insurance premiums on high-risk policies, like where the person is a likely abuser) any more than buying the life insurance policy from someone with terminal cancer. Because the NGO only fulfills the murder if some restraining order is violated, they aren't the agents of the life insurance policy becoming effective, the violation of that restraining order is what killed the abuser, the NGO simply implemented said death upon violation of that restraining order.
But I imagine this would require the abused to be the benefactor of the policy as well as having the right to change the benefactor to the NGO. In other words, the policy would be one that the abuser didn't have any control over. If the policy has to exist in advance for this to not be considered insurance fraud (buying a policy knowing that there is a high likelihood of a very-soon death would be expensive, and not disclosing this would be fraud), this could also be tricky. The abuser would essentially have to have already gotten such a policy before discovering the abuser was actually an abuser (maybe some sort of pre-nup type thing).
This model would also work (albeit very slowly over time) because the NGO would profit whenever the abuser dies, even if they never violate the restraining order. If they have a heart attack or get murdered by their current spouse (who they are certainly also abusing), or just walk in front of a bus one day, they die, the policy pays out to the beneficiary, the NGO.
Aside from all that, the NGO could potentially profit from media rights to the story itself. They are not only buying the murder-right to murder the abuser, but the right to profit from any media based on the abusive relationship, the murder itself, etc. Things like selling books, T-shirts, made-for-tv movie scripts, etc. I don't know how much that would be worth, but the underlying idea being that the NGO could find vectors beyond the murder itself to profit from, assuming they buy them at the same time as the murder-right. What else does the abused have of value beyond the murder-right is the question.