It's not really narrowing its scope to Republicans. It's a non-partisan organization with a policy that has bipartisan support. It's just been especially important to talk about Republican support since Republicans have been largely denying climate change up until very recently.

A carbon tax is required to solve climate change (http://news.mit.edu/2016/carbon-tax-stop-using-fossil-fuels-0224) and only government has the power to levy a tax.

I've been volunteering for climate change for over a decade now, and after thoroughly researching the topic, have finally settled on Citizens' Climate Lobby as the best approach to solving the problem. Based on what I've seen, if we could get ~250 active volunteer constituents in at least 2/3rds of Congressional districts, we could pass Carbon Fee & Dividend. There is actually a majority in support now in each Congressional district (http://climatecommunication.yale.edu/visualizations-data/ycom-us-2018/) and each political party (http://climatecommunication.yale.edu/wp-content/uploads/2018/05/Global-Warming-Policy-Politics-March-2018.pdf) for a revenue-neutral carbon tax. More than 2/3rds of Republicans are actually receptive (https://community.citizensclimatelobby.org/wp-content/uploads/2017/11/june-2017-meeting-analysis-1127.pdf), and that's despite the fact that the typical Republican district has just over ~50 active volunteers (though as many as 283 and as few as 2). We probably only need an additional 45k volunteers in targeted districts, and that's on the conservative side, as it could take even fewer.

Lobbying works (https://sociology.stanford.edu/sites/default/files/publications/friends_or_foes-how_social_movement_allies_affect_the_passage_of_legislation_in_the_u._s._congress.pdf), and public opinion matters (https://www.tandfonline.com/doi/abs/10.1080/09644016.2016.1116651). The biggest barrier is probably that people tend to underestimate how popular these policies are (https://earth.stanford.edu/news/public-support-climate-policy-remains-strong) and that prevents them from taking action.