I would suggest you first think about what you want (financial, recognition, etc.) and then have a conversation with the company about it. As you point out, the company has an incentive to keep you happy! So, start with a discussion here.

I personally believe that going to a lawyer at this stage is 1) expensive and 2) unnecessarily confrontational. I understand the argument about understanding your legal options, but at the point where you start to rely on the law, you're entering a contentious negotiation which can be unpleasant and expensive for everyone.

So I'd just say "Hey, Employer, I've put in a lot of my personal time into this project. So I think it's fair that if you want it that I should be recognized in a concrete tangible way since clearly you want it because it adds more value to the business."

And I think depending on the kind of company you work for, I'd ask for additional equity in the company (since you'd be making the company more valuable) plus additional cash (because you were working on this project night-and-day) and some sort of recognition would all be reasonable asks.