Can someone recommend a lightweight, but powerful, PHP framework that I can build a custom web hosting control panel on?
I have looked at CodeIgniter3, but is that a good choice?
Well I would suggest to use Symfony2 which is really great. It's not really the simplest framework but it will be very performant. Another good point in Symfony2 is the integration. You can integrate other existing Symfony2 applications in your own application (e.g. CMS, Wikis, etc.). This will reduce the development work you'll have to do on your own.
What about Yii2? Pretty fresh EasyiiCMS powered by Yii2 framework and you obviously must have been familiar with the grounds of this framework and MVC pattern. Yii2 guide for newbies.
How to Program With Yii2: Using the Advanced Application Template