I had an evil BG/COT and he couldn't greater smite. I wasn't really that bothered, as Greater Smite feats also give +1 STR per level once you hit level 41, but if you want to Greater Smite with COT, you must be Good aligned.
As you're Pal/BG/COT being good will give you the saves bonus from paladin levels and allow you to g. smite, except for areas in game which shift your alignment to evil - in these areas you'll benefit from BG saves bonus instead and lose greater smite, but there are ways to switch back to good (ask in game).
edit- Just re-read your post and realised i hadn't really answered your question.
You will smite as a COT, not a BG (see documentation for details).
It's not actually possible to make a 20/20 bg/cot as you need levels in a non-prestige class to qualify for either.
You don't get to choose which class you take in legendary levels either, those levels aren't really levels in the traditional sense, but they are based on your control class, which is whichever class you have the most levels in by level 40 (or if there's a tie, whichever class you took a level in first).