I think this is your limiting factor here. I have built something similar as part of a wider, integrated intranet system however if your time is short you can't beat using some pre-existing software.
Nothing will beat ProcessWire if you are likely to want to introduce functionality not available in some other system as ProcessWire is one of the few platforms where you can change your mind about something or come up with crazy new ideas and it not be a problem to make larger changes, but if you're likely to only need a normal WIKI type system then you probably won't replicate all the functionality you want in just a few weeks.
That said, I hate that MediaWiki seems to be the main option out there (there are others, but they all come with their own maintenance issues too) as it's a bloated piece of software (so slow and resource hungry for what it does) on an out of date codebase in my opinion. I would love to have the time and money to develop something in PW as coupled with ProCache you'd be onto a winner and have something infinitely more customisable if a lot simpler, but it would be a project that would take several months at least I would think to cover everything (well, the basics) of what something like MediaWiki can do.
Not sure where my train of thought is going, but I guess if you're really tight on time, you should probably invest that in installing, skinning and getting to grips with something that already exists. If the project is longer-term overall (as in you'll be looking after this system for years), then this buys you time to develop something as a version 2 and does you a favour in a way as you'll know what the client does and doesn't use in the system you've installed so you know where to focus your efforts in a custom-built system.
Hope that's of some help anyway.