Think hard about what this will do or not do to your 10 performing full time people. As you note, "it's just not good for the company to keep an under-performing employee around" and it can become quite toxic.

Being able to fire people is a big and critical thing. I'm not saying "fire her to prove that you can do it", but if you can't do it in general, your chance of failing goes up by quite a bit.

Your proposed solution sounds fair. Can you elaborate on what's not sitting well with you?