Accent isn't an issue unless it prevents your listeners from understanding your spoken English. If that's the case, then you can't claim spoken fluency.
"Bilingual" can have both a restricted and an unrestricted meaning. My 16-year-old son is trilingual, but Mandarin Chinese is his best and "native" language; Southern Min (Taiwanese from Fujian Province) is his "mother tongue" because he grew up speaking that every day with his grandmother, uncles, aunts, friends, and mother, but he doesn't use it as much as he uses Mandarin (all school classes are taught in Mandarin); and English is what he uses with me and has since he was born, but it's his third language. For the most part, he sounds like a native speaker of American English, but because he's never lived there, he knows little or nothing about American culture, and I haven't gone out of my way to teach him. He watches and listens to American movies and to American YouTube videos (comedians etc.) and understands some of the jokes, but he's not bi-cultural and not totally bilingual because he doesn't have native fluency: He often has to translate words and ideas from Chinese and I often have to ask Google Translate to express some ideas and English words in Chinese when we talk.
If you can think in English as well as you can think in Swedish, and if you have a deep understanding of the culture as well as the written and spoken language of an Anglophone country -- one that would allow you to be, say, a simultaneous interpreter at the United Nations -- then you can probably claim "native or bilingual proficiency". You'd be able to simultaneously translate current slang and topical references and jokes from English into Swedish or Swedish into English. I knew a young Japanese-American woman about 30 years ago who was bi-cultural and bilingual in Japanese and American English. When she spoke English, she acted like an American woman, but when she spoke Japanese, she changed personalities and acted like a Japanese woman. Quite different.
If you can easily discuss the major and minor topics in your field of work in English in minute detail, just as you would be able to in Swedish (I'm assuming that you can do that in your native language), then you can claim "full professional proficiency". You probably wouldn't need to use an English dictionary any more often than you'd need to use a Swedish dictionary for arcane technical terms in or related to your field. You wouldn't necessarily be able to translate current slang and topical references and jokes from English into Swedish or Swedish into English.
I'm sure that I'm missing something here, but this is my impression of what those terms mean.