If you are (overly) concerned about the particular political leanings of the institution, I would suggest looking for a school that fits your political requirements first, and second look for a program that suits your inclinations. Since it seems to me that a lot of the LL.M benefit is the name of the school and the career opportunities being there may afford you, particularly if you are interested in political work, stay within the top 25-35 schools if possible (though this is less strict with regard to D.C. schools since being there is half the benefit). If you are more interested in political rather than legal work, where ideology might matter more, I would suggest ditching the idea of an LL.M all together and looking into Ph.D programs or at least dual LL.M-MBA programs or something similar that would provide you with more cred politically than an LL.M will with conservatives looking to hire political staff.
As to your direct question, three schools come immediately to mind:
1. George Mason is a reputable institution with a Libertarian political affiliation and a strong Law and Econ faculty, close enough to DC to matter if that is your desired location, far enough away from DC to be less liberal.
2. University of Chicago is, of course, the pinnacle of Libertarian leaning Law & Econ to this day, and is almost always in the top 5-10 schools for Supreme Court clerks and the like.
3. Stanford also is home to the Hoover Institute, a center of conservative intellectualism, and might afford substantial secondary benefits if you can get in (beyond going to Stanford itself), though the student body and geography mean it is a lot less conservative than the other two.
4. quirky 4th option would be to go to LSE or other overseas school with international renown. (in interest of disclosure I did an M.A. there before Law School, but it has hardly hurt me in interviews).
Outside of those three (four), look for schools with names / reputations for conservative leanings: Washington & Lee is close to D.C., so are Virginia and William and Mary, Duke isn't far away. Emory comes to mind as well. While these schools aren't self-consciously "conservative" as Chicago and George Mason tend to be, they are 1. Southern Schools 2. with stellar reputations. Other than that, and dropping in ranking/reputation look at Baylor, SMU, Vanderbilt, maybe Ole' Miss or something - schools with particular ecclesiastical (Loyola, SMU) and conservative leanings with known conservative cache. Effectively, once you get out of the top 30-35 schools you are simply paying for name recognition or location, so if Chicago, Stanford and George Mason are out for whatever reason, go with a school that stands out to other conservatives (I.E., not UC Berkeley, or NYU or something).
If you are dead set on something inside the beltway, you probably won't find anything to suit your particular political ideology (which I gather is rather to the right of AU, GW, and Georgetown; but Catholic might be worth a look) but there is nothing wrong with attending a school with a "Liberal" reputation. You can't really judge an entire faculty by the reputation of the school. Berkeley hired John Yoo, pariah of the left, for example. Anne Coulter went to Michigan Law. Bill O'Reilly has an M.A. from Harvard's Kennedy School of Government. Nothing is really gained from sitting in an echo chamber. I would argue this is a good thing for honing the skills one needs as an attorney in an adversarial judicial system or a politician in an adversarial political system. Going to a less renowned, poorly located, but more conservative school strikes me as a poor career development decision. And you can generally sniff out faculty of similar political disposition to yourself and develop a relationship with a fellow-traveler.
If this doesn't suit you, I would suggest contacting the Federalist Society, who should have some idea of what schools have solid, active Chapters, or looking for particular legal scholars and reaching out to them for suggestions. Contact conservative legal organizations, CATO, Heritage, Focus on the Family, whatever, that fit your perspective and align with the work you want to do in the future and see what they say. See where the lawyers they have on staff, if any, and where they went to school. LL.M programs offer the opportunity to add a distinguished institution to your resume, don't make the mistake of paying a lot of money and forgoing employment opportunities for a program that isn't going to distinguish you beyond a small geographic area. If you can't get into a good program at a solid, well known school, don't pay the money.