For one thing - Please pretty please don't have an influence bar for the East/West/Unafiliated goverments.
Everyone from UFO Aftermath to XCOM2 had these unpersonal, exactly quantified bars that reduce politics and social interaction down to an integer (or Double, perhaps even Real if your naughty).
They feel about as satisfying as chewing stale bread.
Instead of dealing with "THE US" or "CHINA" - which would most certainly would be an open secret 10min after sending the first letter - The Xenonauts would propably employ Agents to approach resources like power companies, Cable companies or universities directly. Just like the spies of east and west.
If you want to focus on the cold war aspect, then the interaction between agents would be paramount. An incriminating report to the Kremlin and suddenly it'll be pretty hard to access any resources from their land - and perhaps get shot at more frequently from local guards. On the other hand, if the xenonauts further the agenda of Agents, they will pass on information pointing out possible alien infiltrators or valuable ressources.
Instead of a number or a %Bar for the "Relation" ,the actions of the faction should tell you clearly enough how you stand with them.
"Show don't tell"
Several Trucks with Marines on your Lawn demanding your surrender for example would indicate that someone is lobbying against that "dangerous communist terror cell"...
Less severe indications would be propaganda, suspicious intel leaks or harassment of your backers.