I haven't come across any organizations that exactly fit your scope, but here are some ideas.
1. You're probably already familiar with the advocacy initiatives of the major U.S. professional organizations for libraries and museums, respectively:
2. Cultural heritage is often embedded in the built environment. With that in mind, you might consider organizations that deal with historic preservation (which, in the U.S., usually refers to protecting "historic" buildings). Historic preservation advocates may not always consider themselves "progressive," but their principles tend to include things like opposing eminent domain and commercial development if built culture is at risk.
Some of the best places to look for historic preservation resources are sites for historic preservation graduate programs. To find lots of these programs, type this (including all punctuation) into a search engine: "historic preservation" site:.edu
Here are three other strong collections of links:
Lawyers' Committee for Cultural Heritage Preservation - not from an educational institution, but provides some links that are international in scope
3. There are also "centers for community development" (or some variation on that phrase) at other educational institutions that publish reports and, in turn, advocate for cultural institutions (often locally as opposed to nationally). Try:
Center for Creative Community Development at Williams, which seeks to analyze the economic impact of cultural organizations (nationwide) on their surrounding communities. Some interesting stuff.