A Swede who knows absolutely nothing about what you're like as a person is not going to help a petition. The things that would support a petition for pardon or expungement or relief would most likely be people who know you personally but would not be expected to automatically support you, who can speak to your character, and who themselves are, for lack of a better phrase, upstanding citizens (while a felony conviction need not exclude someone from that group, and the whole point of the petition is that it doesn't, you know what I mean when I say "upstanding citizens"). For instance, an employer might make a good reference.
Incidentally, further searching turns up this, where the ATF says that it's been delegated the authority to act on firearms relief petitions, but Congress has prohibited it from spending any money on reviewing or acting on petitions. So while the gun control laws allow for relief petitions, the government can't actually do anything one way or the other on them (as that takes money, and they can't spend money like that). So in practice, you'd probably have to get expungement or pardon, from the state where you were convicted of the felony.
To be blunt: The chance that you will get an armed job is very small. If the military is in need of personnel, you might be able to enlist. The odds of a relief petition are low (unless there's a major change in the politics of the matter that gets Congress to let the government start giving them); other than that, pardon and expungement are the only possibilities for a non-government job, and you still have an uphill battle convincing people to take on the liability of having a convicted violent felon on their payroll and issued a gun.
What you should do is start looking at broader career fields. That's a good idea anyways (it's not like anyone's guaranteed a police or security job), and plenty of people find such jobs extremely rewarding; it's also one of the best ways to convince people you're a productive member of society, which is something you'd like to establish.