The alternative approach is to engage lawmakers early in the process, like other industries do. You can bet that movie studios don't hear about copyright legislation in the press and scramble to influence it at the last minute.
I fear that all a "rapid response" system can do is torpedo bad legislation. A longer term approach is needed to get desirable laws passed, such as defining what is or isn't private.
Speaking of the Library of Congress, THOMAS is the system you want for new, pending, and current legislation so things don't happen behind your back. Not only do they publish what passed Congress yesterday and was sent to the president, they also post the floor activity on the House of Representatives in real time, and daily issues of the Congressional Register. They also have a list of all House Bills and Senate Bills, and a page on each bill that tracks its progress as it moves through the legislative process.
With all of the above tools you will be more informed than anybody who watches the news on television. There's a niche here for special interest issue web apps to make sense of this data. Think Big Tobacco, the NRA, and other interest groups aren't either using or looking into using this data to send alerts to people on the ground in their organizations?
A "rapid response" organization should also have its members in touch with their local representatives on a moderate basis - not too much so the staffers set the "obsessive" flag on their communications but not so little that they wonder "who the heck is this guy?!"
 Article I, Section 5, Paragraph 3 of the Constitution of the United States
 https://www.federalregister.gov/agencies/copyright-office-li... (including a history back to 1994)
 http://thomas.loc.gov/home/rss/presentedtopresident.html (along with an RSS feed)
 http://clerk.house.gov/floorsummary/floor.aspx (along with an RSS feed)
 http://thomas.loc.gov/cgi-bin/bdquery/D?d113:11:./list/bss/d... (each bill has its own RSS feed)
Whatever happened with this?
From the IDL website: "When the internet's in danger and we need millions of people to act, the League will ask its members to broadcast an action. (Say, a prominent message asking everyone to call their elected leaders.) With the combined reach of our websites and social networks, we can be massively more effective than any one organization."
I am for a Slow Response Internet Task Force, where every decision is subjected to a poll along with a debate. Then and only then a "Task Force" is created to ensure the reached consensus is applied.
It's a great idea, and awesome achievement to go from nothing to working site in under 24 hours.
I think the slowest part of the entire process for us (the pdftribute team) was getting the domain registered, and the DNS propagated. Find some way round that bit, and it'll be awesome.