I agree with BikePilot, and would add that you should consider the path that takes you to a practice in "constitutional law." If you look at current high-profile conlaw types - Kagan, Olsen, Katyal, Boies, for instance - there are some common themes: top law schools, academia, and government jobs, perhaps top law firms (see Boies). Those are tough paths to follow.
On the other hand, there are plenty of other opportunities in constitutional law - the ACLU, the NRA, the EFF, the Innocence Project, etc., all consistently deal with constitutional issues. So does anybody working capital cases, and many of those are done pro bono. Free speech cases are also fairly easy to come by - all you need is a high school student with an attitude. So, depending on what exactly you had in mind, there are probably opportunities there.
But, as you have observed, most of these jobs don't pay well. In many cases, the paths to get there don't pay well either. Definitely not easy.
But here is a "trick" to consider: Many schools, in particular many of the top schools, have loan repayment programs whereby they help with your student loans if you take a low-paying "public interest" job (redundancy, I know). This could allow you to take a minimum wage position with the ACLU, for instance, without going bankrupt right away. You will still not be living in high style, but you would be doing what you want.
Which leads to the big asterix, of course - law school graduation is a long time away, and many people change their minds during law school regarding their career goals. Heck, most people change careers several times after law school. So be cautious about the eggs and the baskets, and so forth.
But good luck - some of my friends are doing what you would like to do (or other non-money-motivated jobs), and they love it.