If you go that far into Paladin, you might as well single-class Paladin. If you're going STR-heavy, your CHA suffers, so I can only guess you don't care about spellcasting. Yes, a Paladin 7/Bard 5 split has two more smites and four blade flourishes per short rest. Paladin 11 gives you Improved Divine Smite, which adds a half-smite's worth (or a blade flourish's worth) of damage to every attack for free. You also get your third feat (which if you try to finagle with a Paladin 8/Bard 4 split, you end up with the same number of smites as a 12th level Paladin).
On the other hand, playing Bard 10 / Paladin 2 is more like playing "a bard that can smite" than "a paladin with more smite slots". Going STR-heavy is almost a crime with all the extra spellcasting. An early game possibility is Magic Initiate: Druid for Shillelagh - you can use a torch for a smidge of added fire damage. Later on, you may be able to get a special rapier that uses spellcasting stat instead of STR/DEX.
