It has been confirmed that the order is important. You must score kills with primary, secondary, then melee. No mix and match!
I got this solo on Mercury, Terminus Sector, Exterminate Grineer. My loadout was mostly starting equipment: the Braton, MK-1 Kunai, and MK1-Bo. The later you do this with better weapons or mods, the easier it'll be. The key is doing it with enemies tightly grouped together, you don't want to run to another solitary enemy to hit with the Kunai or Bo. Alarms will definitely help keep them together instead of running away for cover, so don't be in a particular hurry to shut them off.
If this isn't working out, an infested invasion along with quicker weapons would do the trick. The infested constantly rush you, so there's no need to chase down enemies. Swords and daggers are your best bet for melee, with dual pistols like the Afuris or strong kunai for secondary.