So, Admiral Akbar has the following
"During an engagement, play this unit from your hand as a participating unit on your side'
My question is can I play this after the edge battle in the action phase or do I have to bring him out before the edge battle?
Actually, the answer is "yes" to both situations. Ackbar comes out in any Action window if you're using that ability. The key part you didn't highlight when quoting his text is the Action: part. That's really what decides when it can work. So look at the charts on pages 30-31 and anywhere you see a dark blue box labeled "Player Actions" is where you can use Ackbar's ability to deploy him, both on your turn and your opponent's. In an engagement, here's when you can play Ackbar;
After the active player declares attackers, but before the opponent declares defenders
After opponent declares defenders, but before the edge battle
After the edge battle is completely resolved, but before the player who won the edge battle makes their first strike
Any moment between strikes thereafter