I think the Culture Minister should have the ability to overide all decisions anywhere, just till the end of the current term.
oops, did I say that out loud???
Seriously, I would not have a problem defining the circumstances in which the Culture Minister would be able to override, similar to the Military Department. Invasion is pretty easy to determine, so some guidelines for the Culture Department do not seem out of line.
As I see it, I have two responsibilities. One, promote the general growth of culture throughout our empire. I do this by recommending cultural buildings to governors/domestic leader, lobbying for culture-enhancing wonders, etc. This I have tried to do through discussion, building awareness, and the occasional poll. This should not, and thus far has not, required any attempt to override, just as the military would not be able to override during peacetime.
Second, take steps to react to urgent situations which affect our cultural borders. As stated, examples are new cities and those where borders are shifting against us. In this case, since I am charged with looking after the cultural borders as a whole, rather than a specific province, I need to be able to step in and change a build queue. Thus far, I have usually been able to do so by request, not override, and usually pay for the rushing within my budget (Memphis, Thebes, etc). However, if we reach a situation where the governor does not accept my request, or is not available to change the build queue, I should have a mechanism to override it, then if necessary escalate to a cabinet vote to decide the issue. I hadn't thought of keeping budget funds in reserve for mid-turn rushes, I will consider this for the future. Otherwise, I would probably request it be taken out of the accumulated funds during the turn, and then repaid out of my next budget, if it were that critical.
Just my opinion, I want to be careful not to hamstring the office, for myself or my successors, but I would not be opposed to better defining a "cultural emergency". Using the parameters of border control of the city radius seems workable to me. If there is a "culture creep" from another city, but it has not yet encroached on a city radius, we would still have time to discuss and build consensus for cultural improvements, and override would not be required, IMHO.
As for the city investigations, I still think that that would be the role of the foreign minister, I would make a request, but he would still conduct the investigation.