Agree with a few others that have suggested bootcamp type classes... I understand you said non-militaristic, but being in the military, I can assure you that many of the "bootcamp" classes are not what you'd find in the military. Again, you'd get stronger and more fit, but it wouldn't necessarily be in the same vein as Crossfit.
OP mentioned some amount of strength training. I can't think, off the top of my head, of a group environment that incorporates any type of HIIT, plus strength training, and is "CrossFit" light. If there is one out there, I'd be interested, not for myself but just for general knowledge. There are gyms that focus on Olympic Weightlifting and if OP wanted to build a better strength base, getting yourself a coach that focuses on this would help get you there.
Agree with heatseeker and want to reiterate, one reason I usually don't hang out on this particular area of the Forum is that I AM a CrossFitter who has seen great gains in strength, comes from a gym where no one has been seriously injured (keep in mind, there are minor injuries just like in every other intense workout environment), and the coaches will make you strip weight the second you show bad form on something heavy. I don't see why EVERY SINGLE Crossfit thread has to degenerate into this argument. "If you don't like it don't read it" right? Well not when those people have to comment on every damn thread. It's not for the sake of discussion at this point - it's for the sake of argument, and it's damn near trolling. I'm not saying it has to be a loving circlejerk, but any topic on Crossfit quickly descends to this and takes away from what the OP was usually asking for in the first place. Annoying as shit. I love MDA and this website changed my life but frankly, I'm glad I didn't start out on the forums, because I would have abandoned ship almost immediately.
~All luck is earned in the end.~