Killing queens is pretty much asking for a third of the planet's population to spontaneously revolt. If you really wanted to genocide Golds without a civil revolt, covertly sterilizing gold queens would be a much cleaner option.
Alternatively, you could simply sterilize gold political opponents, preventing their legislative powers from growing evermore.
Another more far-fetched option is to simultaneously demand sterilization of all existing gold queens and fund Xenogen treatments for awakening childbearing capabilities in regular golds. Nothing might happen overnight, but in the long term the gold political landscape would suffer apocalypse and a better form of government, reflecting the true will of the people, will arise. Preferably in more than 1 party.
...in fact, that might vary well serve as a good neutral option. Gold matriarchy looses, golds in general come out on top (relatively speaking), and Reds get the satisfaction of total victory.