Hawker probably was taking his whole re-built Alpha Galaxy as it was his personal command and frontline galaxies are designed to move. Real question is how well was it equipped? Since the saKhan was in charge of re-building and delegated a lot of the material replacement duties to Labov . . . its questionable if it was more combat capable than Sennet's Beta Galaxy, though I would expect more Crusaders in the ranks than any other Shark formation. Osis shared some information about what had happened, but indications were he was still hiding some of the information since he still was acting as Jaguar Khan (part of Vlad's charges) so Hawker had a clue but not the full picture. Besides, I think Serpent dealt with the HPG pretty quick. Hawker could move Alpha, filled with Crusaders in command positions and thus least likely to challenge, while moving any other galaxy or even cluster opened his actions up to Clan Council consideration or even challenge by another bloodnamed officer. Getting involved against the IS on Huntress would have presented the Wardens & Clan Council of the Sharks a fait accompli that would likely have compelled them to get involved until Hawker was killed.
After Alpha, Beta and the Spina galaxies would have been the 'easiest' to move into place . . . and would have finished everything the IS had to throw with ground forces & warships.
The Heart was made up of what Osis could strip- and it was that, he was breaking the Clan's operating ability by taking warriors from slots- to build any sort of formation with whatever weapons he could get his hands on. The name was for morale and to describe their efforts.
Nothing AFAIK from the legacy challenge would sway Vlad into helping the Jaguars on Huntress. Its not going to kick off a call of 'Crusaders UNITE!' among the Clans, Vlad's speech in the Grand Council pretty well summed it up. It would not be the end of the Clans, merely the end of a Clan proving its failure while the others learned, absorbed and moved on.