|You don't need to be a stick in the mud like Sturm to be a Solamnic Knight, but it sure helps.|
Of course, then I also need to consider what happens when other classes take that Background. I suppose they remain "squires" and serve a support capacity for the Knights.
Option 2, also fairly simple, is to have all Knights be Paladins, but only those that take the Knights of the Sword path get spells. Crown and Rose get other abilities (or maybe give them feat selections) to make up for the lack of spells. But then I need to worry about balancing out spell casting with other abilities.
Option 3 is to just make a new class that fits between Fighter and Paladin. It's the most work for me, but may be the simplest way to go for the players. And in the end, it would probably look a lot like Option #1, just without the need for a specific Background.
Knowing myself, I'll probably go with Option 3.
As I mentioned before, I'll probably be borrowing some ideas from other places. The Sword Coast Adventurer's Guide suggests using the Fighter archetype Purple Dragon Knight for Knights of the Rose. And I agree, it looks good. They have some good abilities to boost allies, which is a bit different from other Fighter types.
For Knights of the Sword, they suggest the Paladin with Oath of the Crown. Again, not bad. If it can be ported onto the Fighter, along with Eldritch Knight style spell-casting from the Paladin list, it would work well.
I'd thought about the Unearthed Arcana Cavalier path for the Knights of the Crown, but while it's got some interesting things, it's very lance combat focused. A Knight of the Crown with a dragonlance would be a force to reckon with using this path, but there's no guarantee in a sandbox version of Krynn that a dragonlance would ever become available. Lances are also less than ideal weapons in many dungeon environments. So I may need to borrow a few ideas here and there and make up my own Knight of the Crown martial archetype.
Oh, and just for the record, Dragonlance Adventures lists Barbarian, Cavalier, Paladin and Monk as available classes in the setting, so I will be allowing the 5E Barbarian, Paladin and Monk as options, and if I don't give the Cavalier powers to Knights of Solamnia, it will be available for Fighters.
Paladins, like Clerics, will need to wait until the gods have been rediscovered before they get all their cool powers, though...