A lot of people are telling you to cut and run. But that's because of the level of drama you are creating about this, which obviously reflects a very high level of discomfort and anxiety. Which makes you a bad fit to be a crisis manager. They need someone to right the ship. It's possible they're in trouble because the last person wasn't honest with management.
Your liability is limited
As a W-2 employee, you don't have personal liability for the results here in a legal sense. The customer cannot come after you personally in any way.
It is a Unity3D project that a customer paid for (with a huge sum of money)
If the project is still buggy in 5 days, the company will lose a potential customer's future contracts, which can be valued at few dozen millions of dollars
Those are a big bag of not your problem/not your job.
It's easy to get a hero complex and think the result is "all on you". But it's not. It rests squarely on the shoulders of others, who have skin in the game: They've invested their capital in this business, and their personal wealth floats or sinks based on their decisions.
As for the impending crisis, you don't know. It was their job to have their finger on the pulse of their project, and be in constant communication with the customer about it. It's quite possible they are already having the "gonna be late" conversation with the customer, and the owners and the customer are sorting this out. Not your problem.
So customer's information could be accessible by Unity (and so, can create lawsuits led by the customer). None of us have Unity Pro (Unity could initiate lawsuits too, I think)
They can't sue you. You're just an employee. You're protected by the corporate veil, you're not a director, officer or high manager, so you don't have the particular kind of fiduciary duty that is sueable. And you're not a lawyer, and you're not privy to the company's legal work product, so you really don't know.
You should say something like "I am not sure we have the correct Unity software license for what we are doing, please have Legal check it" and your work is done.
So it's gonna be late
First, absolutely very first thing, you must report to management that it is late, and a coarse sense of how far behind, vis-a-vis the parts that are your responsibility. It is more important to do this fast, than to thoroughly research it, so spend 1 hour not 2 days writing this report. This is the stuff they need to know to make their business decisions.
Your next task is to formulate a plan to get your parts done ASAP, albeit late. You'll want to develop that in the next day or two tops. Set up a realistic timeline, and put that in another written report. That is your job.
These are CYA reports. They go on paper. Put a copy in your car before you present it to them, and then hand deliver or present to their admin assistant or their manager. Also send an email PDF. They don't need to know about the copy you keep for yourself. That's a "get out of jail free" card in the unlikely event you need it, but I'm hesitant to say that because I don't want you going into a panic. You won't need it.
Not that this is any of your business, but what's really going to happen here is the deadline is going to come and go, and the company will have talks with the customer and try to figure out what to do next and how to best preserve value. Almost certainly, the customer will assent to a pushed finish date. If your company has your estimates on hand, then they can negotiate from a position of strength and confidence, and that is the best shot at salvaging the contract.
This isn't the first late project in software development.