Is it possible to write and defend a dissertation on work for a private company?
There are certainly ways to get a PhD while working on a project supported by a private company.

It will almost certainly require enrolling in a PhD program, doing the required coursework (if any), passing the necessary exams, assembling a committee, proposing the dissertation, then writing and submitting it. There are few credible institutions that will look at a body of work done outside the institution, go "Well, that's it then." and award you a doctorate.

You would be well served in talking to potential advisors and chairs within the department about their feelings on the matter, as well as your current supervisors - they may not view a PhD earned while working for them as the same as an external PhD ("If we hire you, we get new expertise") so it might not actually resolve your career ceiling problem to your satisfaction.

But in principle, there's nothing prohibited (barring a university's guidelines) in working on research driven by private industry. Though additional complexities may arise - what if the company doesn't want you to publish one of your dissertation papers, for example? Get everything settled, preferably in writing, beforehand. And recognize that working full time and getting a PhD is a tough road.