You do get a Smilodon. But, its medium until you PC reaches Level 7 in terms of Druid Levels.
The Bloodrider is considered, for animal companions, to have a Druid Level equal to Animal Companion - 4. The Boon Companion Feat (which you can grab at Level 5) gives you +4 levels, up to your Character level.
For some context - the Animal Domain from a cleric makes you a druid -3 levels. The Ranger is Druid -3 Levels, the Paladin/Sohei/Cavelier/Hunter is equal to your Druid Level, in terms of animal Companion.
In any case, the Bloodrider starts with the Smiledon no matter what, but the Smiledon is considered Medium.
That being said, the Bloodrider does give a few perks that make its animal companion a bit better.
First, if you mount the thing, it goes faster. And if you rage, it gets a +1 morale boost to attack and damage (as long as you are mounted).
Its supposed to give you other benefits at level 9, but I don’t know if those are in the game.
Right now, I’m only level 6 with my Halfling Bloodrider. I decided that I didn’t want to wait, and since the Bloodrage just gives +2 Morale Bonus to attack and Damage, I just went with my midget rager. Personally, I’m looking forward to Level 7, when the Smilodon gets Pounce. I have no clue how that will impact my midget rager.
Oh, for bonus points, Halflings gets bonuses to mobility checks, so, I will take Mounted Combat… eventually.
Two-Weapon Fighting is a real Feat Sink. As is Boon Companions. But you do get a bonus feat at 6th level, and 12th and 18th… I think.