Seriously, if the opposition party isn't winning, perhaps lurch to the left instead of doubling down on conservatism?
Yeah this is the only real answer, I've long said if the right wants to win, they need to move left, a party that's neoliberal on economics, and comes out in favor of moderately pro choice and pro gun control positions, might make some inroads among moderates on the left. Maybe even a rebranding of the name entirely something like "Liberal Conservative party," "Liberal Party," "Liberal Democratic Party" all could be good names. If you want to narrow in on a specific type of moderate conservatism (i.e. ignore the guidelines earlier in my post) then "One Nation Tory Party" or "Rockefeller Republican Party" could also work.
Not sure how many moderate Labor voters that would pull over without alienating a large chunk of the party. Which would maybe put them in the middle of a three-party system (likely to end up as the UK's edition of your third name, albeit perhaps with some relative success if they can hold onto "main street Feds"), but I doubt it's effectiveness at winning a majority in a two-party system.
See, in the pre-Trump era, lots of Feds would be defined in RL terms as "moderate" or "libertarian," with only a few as "hard right" and some more in between/idiosyncratic. Post-Trump what is considered "hard right" is more mainstream, secular, and edgy so it's much more common on the forum than the likes of Classic/Ben/JCL were during the mid-2010s; meanwhile, libertarianism has withered away even more than it already was (hi) and the wishy-washy of the moderates either turned hot or cold when the mask came off US political discourse.
TL;DR: there is a wider gulf in the factions, this wouldn't work.
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