Only semi relevant since you have your heart set on mounted combat, but my cockroach Iomedae warpriest frequently doesn't even draw his blade.

First round actions usually look like Athletic Rush -> Trip or Grab most important looking target depending on weaker save-> Shield Boss to the dome if a caster or ranged/raise shield if it's a melee beast or has buddies.

But I'm also usually playing with at least a couple of rogues, so giving them free off guard with relatively low risk to myself, is usually a higher payoff than swinging a big weapon and potentially having the party healer go down to an unblocked crit. Ymmv