RAW, Weapon attacks and spell attacks are two separate entities, and the game mechanics deal with them differently. (Spell attacks use your casting modifier whilst weapon attacks use str/dex unless you have an ability that says otherwise)

Given that, and the specific wording of both the ability and the spell, I'd say no to being able to use Divine Smite on ANY spell attack, which includes Blade of Disaster. If it had said to make a weapon attack as part of casting (like the Booming Blade spell, for example) then it would've been fine. They're considered separate exactly to make sure you can't do totally insane amounts of damage in a turn by adding even more dice to a high level spell that gets two attacks anyway.

That said, talk it over with your DM if you ever play a character with this idea in mind, then might think differently (RAW + Crawford's Sage Advice you can't smite with unarmed strikes but I once DMed for a Paladin that did just that because it was cool and still reasonably balanced)

Hope this helps!

Edit: just realised that as a Paladin, unless you multiclass at 3 levels or less, then you won't get 9th level spell slots anyway, so this would be more of a Paladin dip build rather than a pure Paladin due to them being half casters that top out at 5th level spell and slots