The onboard computer does not even know about the bounty before you check for it.
Guys, bounties are an online database and kills are recorded as they happen. I appreciate trying to think of explanations but this is clear. If you have any other insight to offer - or better yet an official source - please do. I will pop over and check the design discussion archives see if there is anything there.
Your logic is fine, but it's just not how it is in game. I don't think FD have explicitly said the bounties are stored on your on-board computer only; and are lost if you get destroyed before you can get somewhere to claim them; but the way they have implemented it, exactly the same as exploration data I think you have to presume that's what they meant.
The information about the bounty is transmitted to you via an interstellar communication network, they have said this much; what they haven't said is that the proof that you killed the wanted pilot, which is what you need to claim the bounty, is transmitted via the same network. You're assuming that because you can find out if a ship is wanted remotely you can use that same network to confirm to the authorities that you have indeed killed it, but even if logically you have a point, FD have never confirmed this and it doesn't match what actually happens in-game.
Perhaps for an in-game explanation, the authorities have decided that network communications are too easy to falsify, they want to see some kind of black-box information on your ship which can't be falsified (because reasons) before they'll pay you for your kill.