That's what we call it in Canada when an elected politician decides to break ties with the party they associated with during the election. They then either become independent or join up with some other party. Not sure if this is a thing in the US as well, but....
How many moderate and reasonable Republicans were elected? Could these people be convinced to vote their conscience and abandon the R for an I or D? How many would it take to swing the power from completely R to something more moderate that would potentially impede the Trump administration?
What would a campaign of encouraging politicians to give up party ties and vote for their country look like?
We call it "crossing the aisle" here, and they can do it without changing parties (though sometimes elected officials do change party, it's pretty rare).
There's nothing to stop a member of either party from voting their conscience and "crossing the aisle" to cast their vote in a way that aligns more with the dominant position of the other party. It's become rarer for them to do that in recent years, especially Republicans. The parties have increasingly diverged on many issues, and while many Republicans may be worried about Trump's presidency, I think their concern is more based on his volatility and lack of experience and what he might do to our alliances, as well as a fear that he may actually do some things that break with Republican positions on social or economic issues. Remember, one of the things that led to Ryan endorsing him was Trump's providing a list of Heritage-Foundation-vetted candidates for supreme court nominations.
Even if a bunch of GOP Senators and congress members joined the Democratic party as an act of protest (and I don't see this happening), they wouldn't start voting in favor of LGBTQ+ rights, women's rights, fewer barriers to voting, pro choice bills, pro environment bills, higher taxes on the wealthy, more spending on education and infrastructure, higher federal minimum wage, more separation of church and state, support for the ACA and so on. Not if they want to be re-elected.
And in any case, most of them presumably believe in their positions as conservatives anyway. Honestly, most in the GOP are pretty aligned with the way Trump seems to be pointing on most issues, barring some of the more extreme measures he's threatened to take against Muslims and Undocumented immigrants, perhaps, and some of his ideas about foreign policy. And nothing is stopping them from voting against any bill Trump supports that violates their conscience, even if they are Republicans.