While riding Merulina, secondary weapons can trigger the melee influence buff. Also tested with non-melee sentinel weapons, but those couldn't proc it.
To reproduce, simply kill enemies while on Merulina in Elite Sanctuary Onslaught.
Tested using a loadout of Yareli, Atamos Incarnon, Furax Wraith Incarnon, and Panzer Vulpaphyla in solo, if any of that is relevant.