As noted, your definition of the scope and complexity of this project is a little vague, but I'll respond with the general observation that larger projects benefit from more "top-down" structure than smaller ones. I suspect that pretty much every PHP developer on the planet started by hacking straight into index.php, then wrote some code for guestbook.php, and so on. Then you realise how much you're repeating yourself and start refactoring to classes and libraries.
Frameworks are the natural next step up from that. The term covers quite a range of products; some that tie you very closely to a specific way of working, and some that are more a library with some loose front control.
I'd advise you to choose a loose MVC framework, which gives you a good structure to work within, but doesn't overly constrain you, and should allow you to use existing libraries. I've not used CakePHP - my experience is with Zend Framework, which I like a lot (not that it's flawless). However, I have worked with another developer to compare the functionality of Cake and ZF, and from what I've seen Cake has many of the strong points that ZF displays. In fact, in many places it almost seems you could convert code from one to another by changing a few classnames.
I suspect Cake's not a bad choice at all, but I can't recommend it as I don't know enough about it. ZF I do know, pretty well now, so I can recommend it - and the docs are now pretty good.
Before you dive into either Cake or ZF, you'll need some understanding of the MVC design pattern. Jason Sweat's book is a good, if slightly dated introduction, and the ZF manual is also pretty strong.
By the way, it's not a choice between "Cake and Pure PHP". Cake (and ZF) are both "Pure PHP". The difference is between "PHP I wrote", and "PHP someone else wrote" (so I didn't have to). From this, the important bit is that you trust the quality of that "someone else's code", which in this case you'll have to do by recommendation and reputation.
But don't just go asking "What's the best PHP framework?" - that's like asking for the best text editor ;) And I'm sure this response (or question) will get tagged 'subjective' in mere moments.