Successfully suing a large corporation is effectively impossible in the United States, regardless of merit. Your only hope is that the case is so egregious that you get a modest settlement offer just to go away, along with the one-in-a-million chance that some superstar lawyer takes your case pro bono. Practically speaking, you will never make it into a courtroom because you will be broke years before the case reaches that point. So the reason to work with counsel before signing the contract is to understand what can be done to avoid the need for litigation later, and the likely consequences of each possible approach. Options may range from filling out the paperwork in a certain way to renegotiating terms to walking away from the acquirer's offer.
If you want non-legal advice, walk away. If you wanted to be working for BigCo, you could have joined up years ago. You didn't. Go find the next thing that's right for you instead.
You don't need a lawyer pro-bono, just one on contingency. If you can survive the motion to dismiss at the outset, you've now entered discovery, and BigCo will now start incurring bigtime legal costs. Settlement offers will be forthcoming. Litigation isn't about winning in the courtroom, it's about inflicting pain until someone says "uncle". (Patent litigation often goes further because the stakes are so high. Your five-figure breach of contract suit, not so much.)
If you want non-legal advice, walk away. If you wanted to be working for BigCo, you could have joined up years ago. You didn't. Go find the next thing that's right for you instead.