Very nicely done, Savage. I agree that handwaves are in order regardless of the route taken by any referee -- or publisher.
1250 - The Aware
The first, as you have noted, is to deal with the Aware in the OTU: it has to change or go away. By the Time of Convergence, it must be different than what it was in TNE. Typically this means Reduction, implying a Showdown in the OTU and some time afterwards for Cleanup, but it depends on one's goals.
Using the K'kree to "clean up the Omnivores' mess" of the Aware is inspired.
The Aware, as such, does not exist in the LTU. If the Aware is smashed by the K'kree, then we have replaced one situation with a perhaps more workable one. How K'kree's newly conquered territory goes back to human hands is another matter entirely.
1200 Onwards - The Empress Wave
The second is in dealing with the Empress Wave in both timelines, which is likely to be disruptive rather than constructive, but THAT also depends on one's goals.
The LTU will be affected by the Wave, but the manner need not be similar to that of the OTU!
1250 Onwards (OTU and LTU) - Post-Third-Imperial Space
The third is in dealing with the former Domains in the OTU. How these are used depends on one's goals.
This is post-LTU, in that history has not been written for this part of the LTU. Therefore, the LTU has room to change, as well. Presumably, its circumstances and the OTU's circumstances could each move towards a gravitational center.
1250 Onwards (OTU and LTU) - Post-Charted Space
The fourth is in dealing with what's left of the other empires. Again, this is purely dependent upon one's goals. Similarly, the LTU and OTU both are open books, and may migrate towards a common gravitational center.
In other words, if there's a central setting goal, then both timelines could slowly be drawn to it as their published timelines are exhausted.