While I personally see the absolute best outcome for Alexios & David Komnenos to become emperor(/s) is for Andronikos I to die before Isaac Angelos is put up as a candidate (a possibility I have explored in one of my previous TLs, one I plan on rebooting at some point in the future), I think there are chances for them to do so afterwards, but before the final Sack of Constantinople.
1. The first of these is a theory that they were conspirators in a plot to eplace Alexios III Angelos with John Komnenos 'the Fat', son of Alexios Axouch, and whose daughter was married to Alexios Komnenos.
2. The second chance I see is when they first take over Trebizond during the reigns of Alexios IV Angelos and Alexios V Doukas Murtzophlos. Say Alexios V holds out longer or drives off the crusaders completely.
Just out of curiosity, which of the two seems more likely?