I would probably write a paragraph or 2 in Swedish explaining that I am bilingual and and can read and write Swedish fluently. This will cover your concern about the company believing that you can not read or write in Swedish but allow you to shine in your native tongue. Alternatively you could write the same letter in both languages. I would probably get some help from someone to proof read to help you improve your non native version though I would still note that I was fluent in both languages.

However if the company will require you to deal primarily(or completely) in your non native language then only use that language when applying.

I would probably write a paragraph or 2 in Swedish explaining that I am bilingual and can read and write Swedish fluently. This will cover your concern about the company believing that you can not read or write in Swedish but allow you to shine in your native tongue. Alternatively you could write the same letter in both languages. I would probably get some help from someone to proof read to help you improve your non native version though I would still note that I was fluent in both languages.

However if the company will require you to deal primarily(or completely) in your non native language then only use that language when applying.