As long as Smite damage isn't doubled on a crit, I don't think it's a problem to declare before or after the attack (assuming the spell isn't wasted on a miss).I'm still confused how this is, in any way shape or form, a problem. The reason it isn't like sneak attack is because it is supposed to consume a resource, you aren't supposed to be able to smite on every single hit you ever make. It also consumes a spell slot because that allows for scaling, you can smite powerful enemies harder than weaker enemies, while sneak attack hits everyone for the same amount every time.
Your problem solely seems to be that the ability was designed not to waste the resource. Which is a good thing, because otherwise, the Divine Smite ability would rarely get used, be a terrible ability and not an iconic one, and all around just be a worse design for what I can only assume is the questionable benefit of easing your personal incredulity towards the order of events.
My experience of Battlemasters is that players will usually spend one on a crit and not even worry about the rider effect but that's not the same as paladins when you are talking about doubling 1d8 to 1d12 vs 3d8 to 8d8. To be honest, if all fighters got one extra weapon die on a crit and Battlemaster dice were not doubled, I could live with that. I would be unhappy if sneak damage isn't doubled on a crit unless doubling sneak damage in the first round becomes an assassin class feature maybe?
Actually, adding Charisma bonus damage when you spend a spell slot might deal with the crit issue if you can live with a paladin who aims for Cha 30.