As a government lawyer who is not employed by the courts or judicial officers, am I prohibited by the Rules of Professional Conduct from engaging in pro bono work? 

No. There may be some limitations on the types of pro bono and access to justice work you may do because of your specific role and any potential conflicts of interest, but government lawyers are not generally prohibited from doing pro bono work (See, Conflict of Interest Rules generally, RPC 1.7, 1.8, 1.9,1.10, and 1.11). Further, some government employers may limit the scope of pro bono work that you might be able to do or otherwise impose restrictions on legal work beyond that of the government client or the government office employer.

Note: Restricting pro bono work of government lawyers may infringe on attorneys’ rights of free speech and association. But see Garcetti v. Caballos 547 U.S. 410 (2006) (holding that when speaking as a public employee and pursuant to official duties, public employees’ speech has no First Amendment protection; however, a citizen who works for the government is still a citizen and, as such, has First Amendment rights that cannot be limited, incidentally or intentionally, by a government employer).