(and it's not like Karabakh is a historical region of Azerbaijan)
Depends on what you mean by "historical". Its historical importance to Azerbaijan may not be as great as Kosovo's importance is to Serbia, but that hardly changes anything. The territory used to be populated by Azeri muslims until late 19th century, when the Russian conquest changed everything, but the Azeri claims to the area remained intact throughout. From its very independence day in 1918 to present day, Azerbaijan laid claims to the area (before Sovietization it also claimed a big chunk of what is today Armenia proper), and held it at first, temporarily losing it to Armenians in 1919-1920, and 1994 to present - but retaining it for the rest of the period.
Isn't wasting money on the military and becoming increasingly isolated only hurting it?
Thanks to oil reserves and extremely strategically valuable location, there is practically no chance of Azerbaijan getting "internationally isolated". As for the military spending per GDP - it's on par with countries such as France and Britain, so nothing to worry about there.
I can't really advise on the best path to take. Russia still has the keys to Armenia's general security, so regular flights to Moscow are a must. But the kind of concessions Russia will want can only come at a price of abandoning lucrative US energy projects, which isn't even worth considering, so in short, IDK.