Been in this situation. Didn't let-go due to emotional reasons. Became much more messy later. Reduced entire team's performance.

Still regretting it after two years. We could have used the budget for a better performer!

It might help to think you're not 'firing' the person for non-performance. Rather you don't have a good fit with him.

There may be other organisations where he may well thrive with more structure and direction (perhaps in services).