So you know where I am coming from, I just finished my first year at my TT position. I was on 2 search committees during that year, so I have 2 seasons of academic job hunting, plus one year on the other side for two departments under my belt. My first year on the market, I had 2 interviews and received no offers. My second year on the market, I went on 10 interviews, turned 1 down, and received 3 written and 1 verbal offer. All of my interviews were at more highly ranked schools in my field. Why the difference?
The (somewhat painful) truth is that I don't know. I changed one letter writer, and polished my materials a little more, but it seems mostly about being in the right place at the right time. My second year on the market, there were more ads that targeted my subfield. That is pretty much the only explanation i can think of.
So that leads me to advice for the National Lab job applicant. First your strengths. If you've been successful at your lab, you have been a PI. You have experience competing for funding, directing research, writing proposals, and planning out a research program. You've been getting at least 2 papers a year, so your CV looks strong. You've been attending 2 or more meetings, so you have been getting your name out there and keeping track of your field. You have a pretty good job, so you can afford to be choosy, both about applying and accepting a position.
You also have some weaknesses. For one thing, you don't have an academic mentor who can show you the ropes (unless you have been keeping in strong contact with your PhD advisor--more on that later). Your postdoc supervisor and/or current supervisor may not be familiar with academic letter writing. You probably don't have an NIH grant (in an NIH field) and definitely not an NSF (in an NSF field). You may not have a track record with external funding agencies. You don't have teaching/mentoring experience. You will probably have to leave all of your current work behind, unlike an academic postdoc.
So what to do?
1. Ask your PhD advisor for advice. Mine was happy to help when he could, even though he is now emeritus, and we weren't in close contact.
2. Have an academic friend look over your application materials. This was BY FAR, the most helpful thing I did.
3. Put word out that you are looking when you are at meetings. I was asked to apply to a couple of places my second year on the market. This is no guarantee, but you may hear of job details (like preferred subfield) that aren't in the published ad.
4. Talk to your academic friends and colleagues about your search, and about making the transition. Most of them will be happy to tell you about their search and give startup advice.
5. If you are applying to R1s, don't worry about your lack of teaching experience DO think about how you will mentor grad students and how you plan to grow your group. DO think about what you would like to teach. You will probably be asked for both at interviews.
6. Make a reasonable list of requirements for your research. You should have a must have section and a really wanted section. This is really, really important. You will have a lot less opportunity to have access to equipment than at a National Lab--most people in academia don't have unused stuff just lying around. It is much, much harder to get equipment buying money in academia. You should also know about how much this will cost. Don't skimp! If you don't ask for what you need, you won't be successful.
7. Keep your search (mostly) to yourself in case it is unsuccessful. Do tell your supervisor--he/she might offer you nice perks to stay, and he/she will certainly be contacted by committees who are interested in you. If for no other reason than to see if there are any strings on your proposed research.