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Other than that, your suggested build only gets a single ASI/Feat, which doesn't even come until late in the game. And you don't have enough levels in Paladin or any other spellcaster to give you enough spell slots to depend on smite damage. Two levels of Paladin is only two 1st level spell slots. That's 2 smites per day.
Level cap of 12 makes a crit smite build very tough, you'll need to sacrifice somewhere. I suggest dropping Barb. Paladin to 5 for Extra Attack and spell slots / Champ to 4 for feat / Bard 3 for more spell slots and the amazing utility Bard brings to a party.
You are effectively a 6th level spell caster (since your 5th level paladin is halved and rounded up) so you get four 1st, three 2nd and three 3rd level spell slots to fuel your smites, plus expertise, jack of all trades, an extra short rest and bard spells that you can cast in combat if you need since you won't be raging.
I'm also gonna go on a bit of a tangent by saying that there's various items which allows you to emulate a lot of class features or desired outcomes. For example, when it comes to having advantage when attacking, there's an invisible polearm giving advantage on attack or a ring giving advantage on attacks but disadvantage on saving throws, disadvantage which can be counteracted by the Undomitable fighter skill.
The great thing about smites is it's a reaction to use, so obviously one would not simply decide to smite a low hp enemy. But there are 3 other characters in a party that can finish off low hp enemies instead of relying on the heavy hitter.
Here's another way to build what OP is looking for without sacrificing as much: Paladin 5 / Barbarian 3 / Champ 4. Extra Attack, Rage, Reckless Attack, Divine Smite and still get two feats. Start with STR 17, take it to 18 with the hag. Great Weapon Master and Polearm Master with both feats. Attack, Extra Attack, Bonus Action Attack from PM (GWM lets you make this attack with +10 damage and you'll have advantage on all of these attacks from Reckless Attack). Suggest not going Berserker barb here since the idea is to use your bonus action for the polearm master attack rather than taking the stacking penalty from Frenzied Strike. Remember, weapon damage is mostly irrelevant in D&D. Most of your damage is going to come from other sources that are applied when you attack.
Or do go Berserker Barb and Frenzied Strike each round. Feats could be ASI STR to 20 and GWM. Suggest turning GWM off when raging if OP plans to keep using Frenzied Strike, as even with advantage, he'll miss more than he hits with GWM also on.
I plan to use Smite as boss killer so number of uses shouldn't be an issue but I will definitely take a better look on Paladin levels. Barbarian post 3 doesn't really offer anything outstanding, this is what made me think of multiclass in the first place.
Lack of feats during leveling is a VERY strong point now that you've mentioned. I have about 3 in my mind I'd like to have - GWM, Savage attack and PM/ASI depending on strongest weapon. I watched this video the other day about Bers/Pal build update, where FS nerf was taken into consideration. Tempting idea ngl but finding powerful two-handed polearm may prove quite difficult task.
Perhaps I'll stick to Bers-Champ combo until I get enough levels, prolly 10-11 and then actually respec. Man this level cap is terrible