From my understanding, bill of rights work is pretty different from federalism/separation of powers work. For the former, the ACLU or a comparable organization, or even a public defender's office (rights 4-8 involve primarily criminal defendant rights) is probably the way to go. You could also look at firms that really emphasize pro bono (see this list: http://www.law.com/jsp/tal/PubArticleTAL.jsp?hubtype=Cover+Story&id=1183107982389;
you might need free registration to access it, but it's a great site so you should take the time); many of them get the more interesting criminal cases for you to work on in moments scattered amidst document review. Also, some firms specializing in media law may get some first amendment-related work.
For the latter, however, you might look towards representing heavily regulated industry--federalism and separation of powers issues, in the cases I've read for administrative and constitutional law, often come from conflict between state & federal law, and conflicts between executive and legislative power. Of course, con law will probably be only a smattering of your work.
I think the ultimate con law experience would be working with the solicitor general's office handling the U.S. position in Supreme Court cases. But these jobs are, shall we say, fairly competitive, and of course don't pay like private practice.
I'm by no means an expert on this subject, just a 2L, but I've thought about it a fair amount myself, and would love to hear from somebody with more real-world experience.
EDIT: Included link