Right now, that label is determined by a primary election choosing the parties' nominees. But, in a system where two Dems could advance to the general, we could see parties using lawsuits and party conventions to enforce their label.
Side note: even if the Supreme Court upholds the lower court decision throwing out the Top Two, the Grange has said they'd consider a statewide non-partisan election initiative.
Anyway, the recent decisions by the Pierce County Democratic and Republican parties relating to how candidates will appear on the IRV ballot next year could give an indication how the parties could live in a Top Two or non-partisan world. Both parties are allowing more than one candidate to appear on the IRV ballot, the Democrats allowing three, Republicans two.
Letter to the TNT (hat tip to Ranked Choice Voting Washington):
IRV is essentially a non-partisan system, as it relates to local parties. Each of them will allow more than one candidate to leave an internal party event (caucuses or a convention) with a nod and a label.
Republicans decided to allow anyone who garners 40 percent-plus of delegate votes at the party’s county convention to run with their brand name. In theory, the party will have a maximum of two candidates for any of the countywide seats. In practice, it will propose one GOP candidate for each race.
The Democratic Party, in contrast, decided to allow an inclusive measure that would allow up to three party candidates per race. In practice, this means that voters will have a chance to decide, based on the merits of each candidate, to actually rank candidates based on their own values and agendas.
So, who's to say that the two parties can't live with Top Two?