What are your ambitions? Why not sell it outright? What would your concern be?

Why not start a company? What would your concern be?

Could you do something different licence the technology out for a year? Could you do something like charge a variable cost to them to keep your upside?

The challenge with your question is that you are asking us to tell you what is right for >>you<<. Unfortunately we don't know enough about you yet to give the right answer.

These platforms may be of interest to you...
https://angel.co/
https://www.fundable.com/
https://www.startengine.com/

You can also apply for a small business loan (depending on your business).
https://www.sba.gov/loanprograms

Lastly, I'd recommend that you don't do any of these things right now.
Try reading these from Brian York (CEO @ Bliss and investor at Bassin Ventures) instead...
https://medium.com/swlh/how-to-raise-your-first-100-000-for-first-time-founders-e9bef553b0e3
https://medium.com/@brianyork/how-to-get-endless-intro-s-to-your-startup-e822b278e956#.yzmonotc0

Gamecenter has extremely poor adoption so I would not recommend you lean on it.

Gamification services in general can be pretty hit or miss. Here are a few for your to explore.
https://badgeville.com/
https://playlyfe.com/
https://www.playbasis.com/

I'd note that these type of leaderboards can often do great things to a product that is already working, but if you are creating something that doesn't already have a bunch of users who really like what you are making... it's a tough road ahead if you outsource (you need to in house the effort).

When we did this for our own company we implemented in house, but we were at a high level of execution and resources (<100 employees).

If you are small I'd recommend http://www.referralcandy.com/. It's a very simple system that you can implement and the payment tiers let you get it fairly "invisible" if you so desire.