I blame the midnight search and you all for finding this thread and staying up way longer than I should have to read it through. And now I have Opinions Post to share...
While I agree with all of those mentioning that Wrath's standing really took a hit with Emperor gone, I'm a bit surprised at the claims that they have no real political standing in the Empire without him. Please bear with me as I go through some of the points (and try to make them as non-headcanon influences as possible)
1) Wrath is the Sith equivalent of nobility
The Empire is, story-wise, is quite big on their lineages, and passing the power / positions through the lineages. While I admit that there is little reference to it further on in the Warrior story - the end-all seems to be that Warrior gets defined by their "Wrath" title alone - the assumption can be made that they, at the very least, have the same standing in the Empire that any non-denounced noble does. That alone grants them some measure of influence, which can be grown further on in the story depending on your choices
Cases in point (hereditary nature of Imperial Sith Lines) :
I'd be happy to hear your opinions on this point - as I said, there is little class-line evidence to support the above. But Wrath certainly was brought up among a higher class than most other Imperial Characters (e.g. Wrath's manner of speech is made out to be quite formal and eloquent in most situations, including those commonly seen as "relaxed")
2) Sith "Right of Conquest" tradition
Sith are also big on the "Right of Conquest" - "to the victor, go the spoils". Just as it sounds like, when it comes to direct confrontation between two (or more) sith, the victor often gains the loser's power base to do with as they please. This is especially evident in the Sith Inquisitors' class line, but is also skimmed over in the Warrior line
Which brings me to the following point - had Warrior not been selected as "Emperor's Wrath" at the beginning of Chapter III, exactly how different would their situation be from the Inquisitor's Chapter III finale? In the end, both executed a Dark Council member, witnessed by other Council members.
True, the SI won Kaggath (and thus had the right to absorb Thanaton's properties, along with the military assets) - but Warrior was succeeding their master (and as per Warrior Ch II, it appears to have the same result). To conclude, had it not been for being chosen as Wrath, Warrior had the right to succeed Baras on the Council, which is something I don't think would just be forgotten (especially if Warrior inherits Baras's power base - which is something that is sadly glossed over in the class line)
Now unto planet/choice specific moments... In which, personally, I think firm Darkside Wrath stands to loose much more than Neutral or Light Wrath
Please pardon if I headcanon a bit in this part - I'll try to mention it where relevant
3) Potential Allies
So... With Emperor gone, Wrath isn't quite as up as they used to be, but personally I'd consider them a sort of "Field Councilperson" - equal but detached from the general Dark Council. Free to do as they please, so long they don't stomp all over other's plans (at which point they have to negotiate, negotiate, and negotiate some more till all parties are satisfied)
Pardon the longs post...