Here’s where I’ll talk about thoughts on the beatdown table. Thanks to a handy pointer, I see the right AC for challenge rating 6 is about AC 15, not 16 or 17. OK. So mote it be.
I’m probably not picking the best tactics, weapons, spells, etc. But I’m going to try.
Without his rage on (but why? OK, OK, that’s one of my categories) he’s throwing down 2 attacks with his battleaxe at 1d8+4. Nothing special here.
While raging, though, if you pull all the stops, you get 3 attacks (a bonus action due to Frenzy) and advantage due to reckless attack, boosting hit chances to nearly 90% vs. AC 15. Damage increases to 1d8+6.
My guy took a shield to hit AC 18. Drop that and pick up a 2H weapon and for one minute you can increase the “10-round burst” to 330, and the 10% loss in AC by dropping the shield probably won’t kill you that much more.
A lot of your powers involve helping others help themselves. You can hand up to 4 d8s to your friends to use for ability checks, attack rolls, damage rolls, or saving throws. You regain this ability every short rest. So this counts for 10-round burst damage, in my book, adding 18 to the total.
Otherwise, the most damaging cantrip is but a 2d4, so that’s not as good as two hits with a rapier. However . . . bards are potent spellcasters, with 4/3/3 1st/2nd/3rd level spell slots. That’s ten spells at one per round, for a minute. Thunderwave (1) and Shatter (2) are the most direct. Thunderwave is 4d8 with a 3rd level slot, as is Shatter. So looking at 3x 4d8, 3x 3d8, and . . . wait a minute,
Sleep basically does 5d8 damage in effect. It’s just it doesn’t do damage. But it’s 5d8 at 1st level, 7d8, and 9d8 with the right spell slots. Heck, it’s not damage, but a sleeping foe isn’t hitting you back either. No saving throw – they just snooze (they can be woken up if their friends decide to do so).
So if you have to kill ’em dead, the noise spells will do actual damage. The 2d8 for a 1st level spell more or less equals your rapier damage, so let’s assume that our bard will throw four turns of rapier attacks and then six shatter stuff. This physical damage output will be about 21d8 for the spells, and 36 HP for physical attacks. 130HP that you deal out, plus 18 more bardic inspiration. That’s 148 HP from our Bard.
However, if he goes all Sleep, all the time, he can cause an average of 306 HP of critters to just fall asleep, no saving throw. Since some might wake up, practically it’ll be less, but this is a darn nasty way to fight if you can pull it off. Especially if you’re fighting small clusters of foes, where you can drop a few low HD critters each round.
One attack, and he won’t get more, but powerful spellcasting, equal to the bard in spell slots (a full caster class). His Sacred Flame cantrip is worth 2d8, and if we assume about a 50% hit rate (1d20+4 vs DC 15) that’s 4.5 damage per turn. His hammer is also 4.5 per turn.
But hooo, boy. Spells. He can open with Call Lightning, for 3d10 (4d10 outdoors) per turn (DEX save, again figure 50% make it for half-damage). This lasts for 100 combat rounds, well outside our 10-round burst consideration. That one spell is thus worth about 82.5 HP per creature targeted, but you only get synergy there if you have more than one target standing shoulder-to-shoulder. Still, in the right circumstances, that one spell can hit for 320+ HP, though low odds of that.
Otherwise, you’re playing the bard’s game with Shatter and Thunderwave for about 130 HP, but again, those are area effect spells.
The Table Thus Far . . .