I'm going to cast a dissenting vote, based on the "obliterating, exterminating war" criterion. I think you can certainly come up with compelling reasons to be in conflict with the fundamentalist "other" nation, but it would always come up well short of obliteration and extermination.
I'm going to be reductive in my definition of what constitute "Western Culture" and define it as capitalist democracy. Any future references to politics/politicians assume a democratic system with capitalist aims.
Ideally, the political leaders follow the will of the people - realistically, they are much more closely aligned to the will of lobbyists and vested interests (including their own self-interest). At best, they can contort the will of some the people to some extent (the classic example being the use of xenophobia as a justification for the erosion of civil liberties - "We're spying on you all for your own safety!")
Before a capitalist democracy embarks upon any campaign of warfare, big or small, it has to address the following:
Is it in the self-interest of political leaders?
Politicians all face one universal truth - their tenure in political office depends on popular support. If they wish to remain a politician, they must remain palatable to the people. Those aligned to a majority party get a bit more leeway to personally espouse an extreme opinion - but are still beholden to the self-interest of their peers. If that extreme opinion threatens the fortunes of the party, they'll be swiftly turfed.
I'd suggest that most modern warfare is conducted for political reasons. In rare cases, there's an ideological reason behind it - but even then, if that ideology is distasteful to the people, it needs to be obscured and a more politically palatable reason given.
In order for political leaders to embark on a campaign of exterminating and obliterating warfare, they would have to believe that doing so is in their best interests, or at the very least, an expenditure of political capital that can be readily compensated for in another policy area.
Is it in the self interest of those who hold the most influence over political leaders?
This point directly relates to the previous one. Political machinations require significant financial support and media reach, and as such, politicians are beholden to those who support them materially. Political donors want some return on investment. That might be broader promulgation of their ideology, but in most cases it's some material gain. They will support policy and action that is either directly or indirectly beneficial to them, or policy/action that is popular because it maintains their influence.
Is it in the interest of the people?
This is again directly tied to the first point. Politicians need public support, and while the populace is arguably the most maleable factor in the democratic process, they become less so in the face of extremist acts done in their name.
People are probably most prone to ideological decision making, but will ultimately act in their own self-interest - with the caveat that many people are incapable of critically analysing the effect of political policy and rely on the media to do so for them - which feeds back into the vested interests in the previous point.
So when we measure against those criteria, I just can't see a situation where a modern (or future) capitalist democracy would ever support the obliteration and extermination of another empire.
The people of an educated, modern society generally care for the welfare of their common man. We even have significant support for animal welfare, and broader concern for the welfare of the environment. For the people to support the extermination of anything, it must be utterly abhorrent, like infectious disease. Given that you've cited shared cultural values, I just can't see enough potential for animosity against the "other".
Even if you could establish a plausible reason why the other empire is incontrovertibly evil, there would be significant opposition to the idea of sending your friends and relatives off to die in large scale warfare. The only way the populace would accept that sort of sacrifice is if there was a compelling argument that not doing so would be worse.
You might be able to find support from political donors who stand to gain from contracts for security, weapons or construction - but again, the scale of "obliterating, exterminating war" becomes an issue. It necessitates unprecedented government control over industry and resources, at which point capitalism and the free market goes out the window. Rather than the overly generous government contracts with minimal oversight that small scale conflict might give rise to, I'd expect the government to enact powers that give them the right to compel any enterprise to cease operations and commit their resources to government mandated activities to aid the war effort, and would expect it to be done without profit.
Without the people or the vested interests, the politicians have no self-interest in starting such a war.
That's why I think the answer is a resounding "you can't". I've completely ignored the idea of global politics and assumed that only the "western culture" and the empire exist in this world, because it becomes even less feasible if there are other states with their own agenda.
Realistically, I think that both nations would gradually become more and more like one another with the advent of globalisation (if that has happened or will happen), and violent conflict would give way to something more "diplomatic" -
where those with the concentration of power and wealth steadily gain control over more power and wealth, without any real regard for cultural or national boundaries.