1. They would have served better by parking up somewhere where they could absorb basic supplies, my guess would be the Chanilaine isles and then send scouting parties ahead.
2. When returning with intel they could allocate their own invasion corridor facing someone without much of a jumpfleet, and militarily distracted. This rules out the Free Worlds League at the time, but it doesnt rule out the Lyrans.
3. Once they have planned everything out, then play to their strengths, attack with overwhelming speed engaging a narrow corridor and cutting deep.
They will need a capital world to take. My guess is Alarion or Coventry, preferably Coventry.
4. The FWL would get involved and shore up the Lyran Alliance, but Hellions move faster than Falcons do, they would reach Coventry before a sufficient defense is mounted, especially if the coordinate their main naval assets.
5. Once the network with the invading clans and try to make old conflicts bygones. They will need the help, and the other clans will need the distraction. There is reason for then to have limited common ground with everyone except Wolf In Exile.