Eu usaria o 2, mas substituindo o "I" por "being". Sou tutor de português e inglês aqui no italki, se precisar de uma força pode chamar! :D

1) I couldn't sleep although I've been very tired. << incorrect I couldn't sleep even though I was very tired. Although I was very tired, I couldn't sleep. 2) I couldn't sleep despite I was very tired. << incorrect I couldn't sleep despite being tired. "despite" is a preposition, so the gerund "being" is required. 3) I couldn't sleep in spite of being very tired. << correct "of" is a preposition, so the gerund "being" is required.

The 3rd is the most correct. #1 would be "...although I was very tired"; #2 should be "...despite being very tired" or "despite the fact that I was very tired" Have a great day!

They are all correct except for a small thing in sentence nr 2. 2) I couldn't sleep despite I was very tired Just change it into this: I couldn't sleep despite that I was very tiered