I don't know what getting the PE is like in other states, but in Maryland you will need four other engineers (at least three of which should have PE certification) to sign your application saying that you're a decent human being. You will also need to document a certain amount of experience working with someone who has a PE (With the EIT I believe you need two years, though I think a post-doc might count as one of those years) and you'll need the signatures of the PE certificate holders who you did your work for.
There are other options as well. For example, many nuclear engineers worked on submarines and aircraft carriers and thus can not talk about what they did for national security reasons. They are allowed to present a project in lieu of documenting what they actually did.
Go to the NCEES website and they'll usually be able to direct you to whatever state regulations are associated with getting a PE registration.
Yes, you can get a PE while working at a national lab, but I don't know how many other PE certificates you may find among the other employees.
I will also point out that the PE is not everything that you might think. What it does is that it is a certificate that makes you personally liable for the success of a technical design. It is a mark of technical experience and ethics that you have learned on the job. You are supposed to be the one to make the final design decisions and it is your name on the line. It is not a guarantee of good judgement. It merely means that you ought to know the limits of your own knowledge and when to recuse yourself from making decisions on things you do not understand. And if your mistake is the result of incompetence, you can be sued.
Nevertheless, I have seen my share of design shortcomings, all made by people with PE certificates.
The reason I got mine was because our company's General Manager wanted lots of PE certificates in the company for bragging rights. My company has no shortage of certified engineers because it is practically a requirement to become a manager in the engineering division. The funny thing is that my employer doesn't insure staff engineers who use their PE certificates on the job. They only insure managerial employees. Thus, my stamp sits unused in my desk drawer.
A PE or a PhD can be a ticket toward management (odd: you'd think it would be the other way around). However I can not stress this enough: Good Technical people tend to be poor managers and Good managers tend not to be good technical people. These are orthogonal skill sets.
Lots of people talk about climbing the management ladder. Some actually go to school for that (the MBA is supposed to be the degree for that purpose). I have to tell you that management is an art. Some are good at it. Many are not. You may find that you enjoy management, or you may find as I have that you enjoy the technical challenges more.
Figure out what you like and pursue it...