The New Republican would be my type of candidate, especially if he supported gun control. This is my map for this scenario:
What I can see happening is generational change pushing the right more socially libertarian, but that is heavily dependent on how you view the demographic shifts relative to the religious right and whether they will be able to maintain their present political influence.
At the same time, you have the economic conditions pushing a lot of voters more towards the center economically as both the right and the left move towards desiring some level of government action and the traditional areas that promoted budget hawk type politics being overwhelmed by demographic change pushing the GOP out of those places (Southern Suburbs) and more into the hinterlands whose penchant for pork barrel and bringing home the bacon dates back to time immemorial (this is also why protectionism has risen in the GOP).
The problem is that libertarianism doesn't mesh well with communitarianism and finding a way to bridge that gap is very difficulty, especially when you have decades of people being conditioned to view themselves as the end all be all and anyone who disagrees is a traitor that needs to be purged. Throw in this the pressure groups and the money and such forth as well as people who make millions and millions more whose livelihoods are in some ways connected to preserving the right as they have interpreted it as being handed down from Reagan, it becomes very difficult to construct a "new fusionism" that instead of combining Social Conservatism with Economic Libertarianism/Neoliberalism (Reagonomics) instead combines some kind of economically nationalist/communitarian right with social libertarianism.
How do you break it, recombine it, protect it from established forces who have a massive monetary interest in preserving the status quo (from news anchors to consultants, to think tanks) and get it to a point where it is a viable political force that answers the economic needs of the people and comports with their socio-cultural norms, without it being blown up along the way by any number of pitfalls along the way?
Is it doable theoretically? Yes, but it is very difficult without radical reform to how politics is financed and money influences it. After that you need a complete replacement of most of the leading political figures on the right, the destruction of and elimination of the DC think thank and consultancy class, and the replacement with competent, qualified and responsible people who understand this new combination and will be able to fill the void without succumbing to the same corruption and extremism we have now.
In Atlasia for instance, the game has existed without much in the way of an Evangelical Right for years, there is no money in politics and there is no consultancy class/think tank culture. Most of what resembled a real life Republican Party failed to survive the Bush years. What was left was that for a long time you had a libertarian Southern region and some social conservative/economic populist Catholics in the Great Lakes making up the large bulk of the two major right of center parties (The RPP and then later the Federalists), and political reality dictating a greater accommodation towards a big tent, towards accepting moderates and cooperating with each other behind a platform and policies that would be competitive in a set of demographics in game on the national level that probably resembles that of New York IRL. Even then, the Atlasian right is still largely pro-gun and pro-life, though unlike RL, these aren't purge level litmus tests.
Seems the necessary ingredients here are:
1. Overwhelming Left and center left dominance
2. Decline of the current core of the RL Republican Party as a force in politics
3. The removal of monetary influence from politics
4. The destruction of the current party establishment and hangers on
5. Possibly the destruction of the Republican Party itself and replacement with something else
6. The formation of a new fusionism by getting two groups that seem on the rise, to somehow get in the same room and work out a set of agreements and that becomes the new platform.
This takes decades to happen, and you are talking possibly eliminating a 160 year old duopoly and replacing it with another.
