It would be plausible enough, but I'd rather see the keys as very special rewards, not simply given by their Council friends. If there would be a long and difficult arc to get there, then why not.
Perhaps a random, tricky mission then? "A an armed convoy of one of the alien-allied countries is transporting goods as well as access key to the alien embassy in the area. You have a possibility to intercept it but be aware - it is defended by regular human military merely following its orders and you will be penalized for any casualties" for example, where hostile defenders are well, alien-tech armed military whom you have to disable non-lethally or pay a hefty price of standing with the council/region on top of losing some points for stealing the cargo of the convoy (simply negative point values for killing your enemies would do).
I don't understand this argument - what you described is 100% political. How can a black ops/terrorist organization be anything else than a political organization? This article looks 100% correct to me.
Ah, you see, the differentiation is as follows - usually, by "political organisation" one understands groups that operate publicly, gathering personnel and making a push toward certain ideology or established interests of said political groups. Black ops is off-the-books, hush hush operations which while often utilized to further certain political agenda but that's merely as a (perhaps intended but still) side effect of their direct action, which usually means eradication of opposing forces, sabotage, illegal acquisition of data or material.
The issue is that otherwise you can claim that
any and
every group and organisation associated with the Council is political as it furthers Council's agenda, even if it's local hot dog stand furthering said agenda by keeping some Council member fed with his favorite hotdogs before he goes to work.
Plus, hey, you wrote that:
There are many reasons why Alien Embassies shouldn't be removable with no limits, mostly because X-Com is not a political organization
which is why people bring that ufopedia entry as X-Com is described as political there.