My humble suggestion is for a breakaway to call itself...
The Traditional Republican Party.
It could caucus with the (cough, cough) GOP, and ideally garner some committee chairmanships.
The name alone would bring in disaffected GOPers.
.
Once the new party is established, we should have a big event where we burn our voter registration cards. Just might confuse enough aging hippies that they join by accident.
Traditional Republican Party.
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I like your idea about caucusing with Republicans. This gives some politicians who would want to move toward us a way to do so. Also, this looks dignified and thoughtful. Others would want to paint this as a movement of sore losers who worship Trump, but this reminds them that this is not a cult of personality, but a reforming movement to restore the best of our heritage of freedom and opportunity.