UK and EU both have agencies complimentary to those I mentioned, as you probably know. I don't know much about their hiring practices but I assume they would be similar.
Anyway, the point isn't so much government work as to say that a good % of chemistry PhDs have jobs that never require them to go into a laboratory. Hell, for that matter, have you considered applying for professorships? You can have students go into the lab for you
In seriousness, one mistake I think a lot of PhD holders make when looking for jobs is excluding themselves from consideration because they don't think their background is a perfect match to what is being advertised. People come out of PhDs with an incredibly narrow set of technical expertise. If you are looking for an opening that needs just that specific set of skills, you will be looking for a while. What recent PhD holders don't realize (I certainly didn't) is that a lot of places hiring PhDs are looking for the problem solving and other skills advertised by the PhD itself, not for someone with a background is some exotic form of LC/MS. It took me a while to learn this on the other side of coin as well - as a hirer. When I first starting hiring people (for lab positions), I agonized that I couldn't find an applicant with the exact technical skills I needed. The point I was missing is that a PhD implies that even if the person doesn't know that exotic form of LC/MS, they will be able to learn it without me. I no longer look for technical backgrounds when I choose candidates. I look for people with the best all-around CV, whether it's the greatest technical match or not. That strategy hasn't led me astray.
This applies even more-so to non-laboratory jobs, where your technical skillset just doens't mean a whole lot. So, I guess my advice is to apply for anything that seems interesting and is a reasonable fit to your background as formulated in broad terms. I mean, don't apply for something if they're looking for someone with a PhD in astrophysics, but you should feel reasonably confident applying for any nonlaboratory position that requires a degree in chemistry, and certainly organic chemistry (whether its synthetic or not). Don't exclude yourself from contention before the hiring manager even has a chance to look at your resume.
(As a personal anecdote, I came out of PhD doing ultrafast spectroscopy of conjugated organometallic complexes; the job I applied for, and got, was in polymer packaging, no spectroscopy, no organometallics. And that was a lab job. But I had a PhD. I adapted.)