No paladin or ranger levels? What about druid?
One of the biggest abilities of the Halfling Outrider class is that levels in it stack with levels in paladin, ranger, and druid for special mounts and animal companions. I think you are going to want to cash in on that somehow so that your mount is tough enough.
Druid 7 can get you in the door, if your dungeon master will let you trade out wild shape for something more appropriate for your character it might work pretty well.
Trade in wildshape for fighter bonus feats like the rogue can trade in sneak attack for fighter bonus feats?