One way or another, there's no way I'll be in this position for 5 years. They won't keep me in it, and I won't stay in it.
The major question is whether I look for an internal transfer (this is a big and generally quite good company) or another job. And if I do look for a transfer, do I wait or start immediately?
I had this same dilemma 5 years ago. I was right out of college and started a new job. I hated the project I was working on (it was one main project at a medium-sized company, so I didn't have much choice) and I dreaded waking up in the morning.
I lasted about 8 months. I used my lunch hour to send out my resume and find another job. It was the best decision I could have made. My new job was much better and it lasted until a year ago.
It's more than just your job. When you enjoy your work, your entire outlook on life is better. During that 8 months, I didn't work on any side projects, because I was so depressed. The plus side is I never went out or bought anything and I got myself out of debt.
"The major question is whether I look for an internal transfer (this is a big and generally quite good company) or another job. And if I do look for a transfer, do I wait or start immediately?"
It's an option, you can try it. But, your boss may not put in a good word for you, which will make things difficult.
Can you imagine being in the same position for 5 years?
I would look for another job.