Here me out,

Titus Mede II is a warrior-king whose aware that he's become a political liability to the Empire. Even if he made the right decision in appeasing the Thalmor while they rebuild their territory, everyone believes he surrendered Tiber Septim's territory to elves and blasphemed against the God of Humanity. He can't rally the troops around the Empire anymore since he's gone from being a badass warrior king who leads from the front to being another word for craven. His every decision is suspect and the Nords think he's a weakling.

So what's the best solution if not to kill himself? It fits with the Roman ethic of the Cyrodiil Empire and we see that Titus Mede is completely alright with the Dark Brotherhood killing him. He's resigned to the inevitable and seems positively happy about the development. Mede ordering the execution of his "killer" may actually just be him tidying up loose ends.

Titus Mede II being assassinated by the Dark Brotherhood allows him to end his reign in a manner that he can prepare for with a chosen heir ready to take up the cause. Furthermore, it allows the majority of the Empire's wrath against him to die with him. Titus Mede II the appeaser being killed allows a Mede heir (or even a chosen general) to be the Winston Churchhill they need against the Thalmor. A uniter rather than a divider.

Hell, the Empire might even be able to mend fences with Hammerfell.

Did he order it? No. I don't think so.

Was he alright with it? Yes. I do think so.

You've got the rationale right, Titus knows he is a liability to the Empire, and he knows his credibility has been shot to pieces by the WGC.
But I do not think he organized or ordered it, otherwise he wouldn't want you to kill his contract holder, loose ends or not. And Motierre says there is to be a chance in policy.

I think Mede stayed Emperor because he knew the Empire needed some time to recover, if he died right after the WGC the Empire might just go to pieces, right back to succession crisis. Waiting a while for the Empire to put itself together meant that if someone has the political juice to kill him, they might have the clout to replace him.

I don't agree with Titus' choices, but this seems the best reading I can think of.

I've been considering something similar. Mede always seemed to be planning ahead, and is clearly far more intelligent than most gave him credit for. His advisers recommended against rejecting the original Thalmor ultimatum, but he did it anyway. His generals recommended against abandoning the Imperial City, but he did it anyway. He then led what some would call a desperate attack on the Dominion which saw the entirety of their army in Cyrodiil slaughtered.

Even when it comes to his death, he came to Skyrim against the advise of the Penitus Oculatus. He stands there and lets you kill him (i don't buy any of this 'using the Voice of the Empire' to convince you not to crap) despite the fact that it seems to make little sense to anyone around him.

I'm not sure what his game is/was, but Mede seemed to be playing a much bigger game than we see. Considering the access the Emperors have to the Elder Scrolls, and the fact that the only other Emperor we have any experience with took a vested interest in prophesy, it's possible Mede may have been up to something...

That said, do i think he ordered his won death? No, but he sure as hell let it happen.

Definitely not.

Mede's made some poor choices but I don't think he's that stupid. No good can come of creating chaos among the leadership. If he wanted to step down he could name his successor and do so, with far less dire consequences.

Having his death be an orchestrated murder of not only himself but his cousin in the midst of a civil war in such a way that it further fractures relations between skyrim and cyrodiil is so ridiculously foolhardy that it had to have come from the council, not mede.

Not necessarily. There are some paradigms to be drawn towards Mede's actions and the socio-political nature of Cyrodiil. He waged war like a Colovian, and then -Tried- to be diplomatic like a Nibean. In the latter case, we can't really say whether he succeeded or failed, as we're not really sure about the attitudes of Cyrodiil at present.

Still, its safe to say that he was losing political support, and had he just stepped down, that lack of support would almost certainly carry on to his successor. The sins of the father and all that. By dying, he takes his sins with him, ensuring whatever successor replaces him starts with a blank slate. And an army sitting on the Dominions border.

The Dark Brotherhood is also the perfect people for assassinating the Emperor. They're linked to no factions, no one likes them, and they're incredibly anonymous. As a result, there's no way any chaos can be generated from his death. However, people's suspicions about who hired the Emperor could be quite interesting if harnessed. People might think he was killed by the Thalmor or other parties.