When a folks started opening their boxes and playing against the Syndicate for the first time, there were lots of complaints of stalemating or games taking way too long. But when questioned further, these issues largely came from bad strategy in the begining. In that case, most people were not even thinking about getting the flagship out early to kill it and just running themselves in circles.
Omega is, in my opinion, the hardest of the Opposition forces. Your first 2 rounds are really going to set the pace for the game, and early misteps will only compound later.
What did your opening setup look like? This is key. You have to deal with Alphas, at least get their shields down a little. The more you allow those Alphas to live and grow, the worse you will be once Omega shows up. You have no choice there. You have to stall the flagship while you find a way to manage the Alphas. The nice thing is that even as Alphas absorb the hunters and gatherers, those little guys become easiler to take out, which will buy you time preventing an overrun in the sector with the starting mission.
If you were playing with Elite Mode Icons, its near impossible to stall Omega for more than 2-3 rounds at most. But you know that Omega builds very quick, which means if you CANT deal with Alphas, then you need to switch into the "I better just build faster than it does" mind set. You took Vigilance and Basilieos. THose are both incredibly fast building ships. If any SF ship can catch up to Omega, its those two, so I am a little perplexed by just how few energy you actually manage to build. Keep those decks smaller and try to not buy as much useless stuff for your ships (unless its because you really dont want it to flip, which will happen granted). Going up against Omega, speed is the strategy. Things like Desperate Measures and cards with will allow you to nail Omega for Energy damage are very usefull to bring his shields down.
You have Omega parked in the Trade Outpost. The Wastes would have been a far better choice to flip. Why? Less cards flipping and scrapping in the Aftermath! He gains +1/+1 for each ship scrapped over the fleet limit. You won't play every game with the wastes granted, but that would have been a far better option to know before you even got to the first [T] phase.
Hope some of that helps a little. The thing is, if Omega spawns too early and gets way too many Alphas, or eats a huge Alpha, you're in trouble. So plan accordingly. "Stalemates", though not quite an accurate term, are a result of poor strategy in the early game. Adjust your first few rounds and you will see the game going much smoother. Its the same way for any Opposition.
Edit: I have a big session report I wrote on the forums about a 2 player game I had against Elite Mode Omega. Our starting ship flopp was awful! Maybe reading that will give you ideas since we ended that in a win