>> The source of denying weapons to the Christian Militia is not the Kurds
An important clarification.
The KRG Kurds are wary about an Assyrian independence/authonomy movement ouside the KRG framework... as it affects important areas that the KRG wants for Kurdistan.
The protective role the KRG has assumed through sheltering Christian refugees and caring for Churches etc. is also to win Christian support for the notion that under Kurdish rule the Christians are safe. But they have to convince the Assyrians to overcome the bloody historical experience between Kurds and Christians. Also the KRG wants to dispel any notion of independent Assyrian movement.
For this reason (along with pragmatic concerns) the Kurdish Regional Government prefers all minorities to serve within the Peshmerga and is critical of ethnic-religious militia on their own.
There also have been direct clashes between Christians and Kurds in Syria:
http://hotair.com/archives/2016/01/13/kurds-attack-assyrian-christian-village-in-northern-syria/
http://www.breitbart.com/national-security/2016/04/25/christian-assyrian-clash-kurds-syria/