Upgrading a mount: how should I proceed?

One of my players started at level 1 with a Riding Horse through their backstory, and they knew that they were going to pick the Cavalier Fighter subclass so I thought it would make sense anyway.

But we're nearing level 6 and stakes are becoming higher (running SKT so big statblocks around) and their ride, even with a decent barding, is only 1 hit away from the grave.

I wanted to find a way to upgrade their mount. One of the things I thought of included making their horse a sort of familiar he can summon through Find Steed eventually, but I was wondering anyway if upgrading a Riding Horse to a Warhorse would make sense gameplay-wise, and how to incorporate an appealing side quest to that.