I believe you can operate as a sole proprietor. As long as you keep your book-keeping under control. Check out https://www.waveapps.com as a financial tool. (I'm not associated to them in any way, if not you'd see a referral link :P )
While I agree it probably would be a bit undertaking, my personal take on something like this is this: Firstly, pick a framework to build on top of something like Wordpress, or (my personal pick) Drupal.
If you do this in something like Drupal 8 which is how i would do it. Then you have knocked out a lot of the easy stuff like User management, permissions, etc. You also could lean on things like the Commerce modules to take care of all of the payment tracking.
As far as actually serving the advertisements, using an RestFUL callback that returns the completed advertisement along with the correct referral codes should.
In regards to tracking the Traffic there is another OpenSource project called OpenWebAnalytics, http://www.openwebanalytics.com/ it has the capability of tracking traffic very similar to how Google Analytics works. It is written in PHP, and open source so you could alter it to suite the needs. I believe it has a built in pruning process(dumping old data).
So really after all of that its a matter of tying all of the pieces together, Im someone who just by chance is good at this, checkout my blog http://www.wembassy.com/blog for some of the things i have written and read up on. Maybe we can connect and help each other out.
PureVPN and nord have similar commissions. I don't know about tunnelbear.
However, I think Pure VPN might have a better one because there's so much hype around it and many have heard of it.
But, it also depends on your audience. Are they tech savvy or just scared people...
Bitly is still free though. Maybe you did not scroll down far enough (https://bitly.com/pages/pricing) or maybe you reached your limit?
If you use Wordpress then look into using a plugin instead like pretty links.