I went part-time the first three years, and I had time for Law Review and Moot Court. It sucks, but it can be done.
The biggest choice you will have to make is your last summer of part-time. If you want a firm job and land an SA position, then you either need to take a leave of absence or quit your job.
As far as I could tell from my experiences and those of my classmates, employers rarely knew or cared if you went part- or full-time.
Going part-time can cost a lot more. Because of paying for all 90 credits I have taken (instead of the 72 the full-timers pay for), paying fees for an additional couple of semesters, and tuition/fee increases, I paid 33% more for my education than I would have if I went full-time from the start. At my school, that wasn't an astronomical amount of money, but if you go to a school where you're paying $20k, $30k, or more per year, it's something to consider.
I know Georgia has an exception where if you transfer with your current job to Georgia, then you get in-state tuition from day one. This would apply at a school like GSU. I don't know how working from home counts, and I don't know how many states offer that, but it's worth looking into. That being said, (and I could be completely off the mark here), don't you have a shot at Georgetown/Fordham? Both are pretty pricey (assuming no scholarships, which, as far as I know is still quite often the case for PT students), but they might provide you better options than many other part-time programs.