Yes, I forgot the Japanese HMG.
Some warlords did use Italian weapons, however, I don't know much about them. I only know they seem to have some 6.5mm Mannlicher-Carcano rifles. Chinese army in WWII also had some Fiat HMGs(both water-cooled and air-cooled), but I'm not sure whether these guns were bought during civil war period or were bought from Mussolini in 1930s.
Italian rifles were cheap at that time. I remember seeing a table of the prices of different kinds of rifles in China from 1920s. Chinese and Japanese rifles were the cheapest, and next, the Italian ones. The price of one Italian rifle was only about 1/3 of the price of a Mauser M1896(the box cannon). Now you can understand why "pistol troop"(手枪队) or "fully armed with box cannon" meant elite troops at that time.
I'm not an expert on smallarms...
P.S. Do you have some further information about the Yunnan army's French tanks you mentioned in the thread "Armaments of China and Siam to 1949 Part 2", especially photos?