Azerbaijan is not actively on West's radar right now due to the small size of conflict with Armenia and the latter's flirting with the Ruskies. But if they choose to go down that line, Armenia's Christian card will very well be used by the Church lobby in the West. They're not even big enough like Pak and Turkey to survive for long. India has a good trade with these bhaijaans though XD
Our relations with Azerbaijan have only recently been impacted. The good trade is due to oil & gas and the post-Soviet sentiment. The moment Bhaijans start messing around, we will switch. Armenia sadly got a very bad deal despite its loyalty to the USSR and later Russia. While Azerbaijan & Turkey bonded over being Turkic & Muslim (Pakistan was not in the picture back then), Armenia was surrounded by only 2 friends, Iran and Georgia. Most Armenian men have migrated to Russia for work and a lot of them work there. Some villages don't have men at all as almost all of them migrated to Russia for work after USSR broke. Also, their division has been such that they offer no strategic value to others, unlike Baku that has a Caspian Sea coastline and lots of oil.
This has put Armenians in a bad position, geopolitically speaking. While the India-Gulf-Europe corridor has Armenia involved in it, guarding it against Turkey & Azerbaijan would be a problem in the future. Russia is involved in a complex foursome with Turks, Azeris, and Armenians right now and leaning more towards the Turkic lot due to their strategic benefits. Since this is a more West-oriented project, Russia will be indifferent to it or worse, hostile. Therefore, Armenians & Georgians will have to do their nut to guard this trade corridor from any disruptions.