Well, I liked constitutional issues too, and like a lot of people that was a big interest of mine. The problem is that almost no one does that. You may get to do it some as a criminal lawyer, but criminal defense at least was pretty miserable to my experience. To more generally answer your question, it's an over-saturated profession whose governing/organizing bodies would be hard-pressed to give less of a shit about what happens to the rank and file. Don't have a job? Oh well, still have to pay your bar dues and student loans. Meanwhile the state bar likes to have its mid-year meetings in Europe, the implication being that if you can't afford to take 10+ days off work and put yourself up over there, they don't really give a shit about hearing from you. Doing your continuing education online too easy? That's fine, we'll add a requirement that 1/4 of them (per year) have to be "interactive," which of course helps the monopolistic course providers that charge 10 times as much. I could go on, but you get the idea.
It may not be as heady as Brown v. Board, but is practicing law while working for the government a sort of constitutional law?
Not usually. It's typically little different from any other practice, just you happen to have the government as a client. It can sometimes mean additional ethical issues or whatever, but more than anything it'll depend on your area.