How can I suggest this to him as a peer giving advice rather than as an unsatisfied team lead trying to send him away?
You can't and it's foolish to try. You're not a peer. You're this guy's manager. You need to manage him. Whatever you tell him will always be received as coming from the manager, no matter what you do. But that's not a problem, because it means you can actually do more than just drop hints.
I can not bring myself to suggest that because it can seem like I am kicking him out or something
The way to do that is to actually have discussion with him. You say you want to drop some hints or suggestions that he should look into moving on. But that's dancing around the problem and you need to address it directly:
You've mentioned before that you're not excited about the kind of work our team does. While I don't expect people on our team to love every aspect of their job, it's a bit concerning that you don't seem to be interested in our core tasks at all. And if I'm being frank it's starting to show in your work. Since the nature of our work isn't likely to change I think it makes sense to start thinking of alternatives. You may not know this but we do try to make it possible for employees to move to different teams or branches of the company if they want to change their focus. You spoke before about wanting to move more toward a [...] position and we do have teams working with that. If you want I can look into setting up a meeting with [other team leads]. But if you don't think that's a suitable option I do need you to start taking this work seriously. I'd expect [X, Y and Z] from someone in your position and right now I'm not seeing that. You may not think this work is that exciting but I do need to have a team that gets it done and can commit to the job. Can you do that?
That's a bit of a long script but it touches on all the aspects you want to cover in the opener to that conversation. How it plays out depends on the employee. You should also give him time to make up his mind and not press for an immediate decision but you do need to get him to make a decision. Either he decides to move on internally, he commits to the job as it is, or you'll have to start transitioning him out. That's how you manage someone.
Whatever you do, make sure that moving to another team is indeed possible and encouraged. Ensure that the teams you suggest or that he would likely want to join are willing and able to take this guy on. And be absolutely sure that you're honest about this guy's skills and performance. Don't ever play hot potato with a bad employee.