This question is slightly removed from the "government - singular or plural" question, though the two are connected.
In BE, we are happy to consider government as singular or plural - as explained above. etb has explained the rationale for our choice.
If we find ourselves needing a relative pronoun for government, it depends on the same rationale. Speaking of government as a set of individuals, we use
who; speaking of government as an impersonal entity, we use
which.
In AE, as I understand the opinions expressed here, government is singular and therefore requires
which, not
who.
I can't imagine either AE or BE tolerating:
The government, who has a good reputation, has not solved the problem yet.
So the answer to the post #1 question, as amplified in post #3, is straightforward: the sentence considers government to be singular - use
which, not
who 