Post may refer to the specific job of a diplomat within a diplomatic organisation or embassy.
post = a job in a company or organization
In the context of your question, post refers to an embassy or diplomatic office in whichever countries are referred to. An example of such usage is in:
For the first time, China now has more diplomatic posts around the world than any other country, an Australian think tank says.
According to the Lowy Institute, China overtook the US in 2019, with 276 embassies and other representative offices globally.
That's three more posts than the US - France, Japan and Russia are in the next spots.
The UK comes in at number 11, down two ranks from 2016.
The opinions expressed to Washington are likely to be the considered and co-ordinated opinion (or requests for action or decisions from Washington) of the post rather than of individuals within it. Individuals may change at short notice but the entity of the post remains and it is it rather than the people within it that corresponds with Washington.
The term seems to have evaded dictionary definition but is widely used. Here is another authoritative example:
China has overtaken the US as the country with the largest diplomatic network, with 276 diplomatic posts around the world — three more than Washington — and ahead of third-ranked France. Beijing ranked behind both countries when the index was first published three years ago.
The number of UK embassies has remained steady at 149 since the 2016 rankings were released. But the government has closed or downgraded 11 consulates and diplomatic offices, including posts in St Petersburg, Russia and Alexandria, Egypt, according to the Lowy index.
“While the stakes of Brexit are highest for the UK, it has been slow off the mark in preparing its diplomats for Brexit,” said Bonnie Bley, research fellow at Lowy.
“By contrast, Ireland and the Netherlands have taken determined steps to boost their networks as part of their Brexit strategies, adding eight and seven posts respectively.”
Ireland is ranked 40th on the index with 87 overseas posts but experienced the largest boost to its diplomatic footprint, rising from 43rd in 2017. Its foreign minister has attributed this expansion to its “Brexit strategy”.