Among the putatively Republican interest groups that would seem to have the least pull with the GOP, I would include the Log Cabin Republicans and the Republicans for Environmental Protection. As Andy Revkin notes in his interview with the leader of the latter,
This group, while holding to traditional conservative values, has positions on energy and the environment that are substantially in sync with those of many Democrats and independents.
Now I would argue that the guys who have political juice within the Republican party are the traditional hook and bullet groups. They're the ones that kept the Bush Administration from completely obliterating wildlife from the Western landscape in the 2000s. (Okay, some vigilant environmental groups played a role, too.) These hunters and anglers are also worried about climate change. Of course, at the end of the day, they have as much sway over the GOP's stance on global warming as the Republicans for Environmental Protection. But given the significant membership rolls of the Hook & Bullet groups, I'd say their collective voice is the one to pay attention to when it comes to conservatives and climate change.