I mean, sure… I won’t go super in-depth so this may be ultimately unsatisfactory, but let’s give it a shot.
The main concern is that the Barb 6 gets their 5th level scaling via Extra Attack while the Fighter 3 / Bard 3 does not get any 5th level class feature.
Fighter 3 gets some Weapon Masteries just like the Barb, so we can assume they both pick the same weapon, to attempt to normalize things. Either they both pick heavy weapons, or both do TWF or whatever. I don’t have much preference. Let’s say Greatsword for 2d6 and Graze.
The Fighter also gets a Fighting Style which the Barb lacks. Let’s say we pick GWF to keep going down the 2-handed route.
The important point to ponder is what does Bard 3 gets us. It gets us spells of levels 0-2, and the ability to allocate our concentration slot… if we start with cantrips, since when comparing martials we often like to think of the sustained combat angle, we can highlight Bladeward and True Strike.
Bladeward can be a fine way to use our concentration without expending resources (besides the Action to cast it). Depending how your DM handles cantrips that last a minute, you may be allowed to keep it always on (i.e., always cast it once a minute when walking in a dungeon). Some tables handle this by saying you have 1d10 rounds left on the spell at start of combat. If that’s not allowed, then we have to consider if getting +1d4 of AC is a valuable enough use of an Action.
True Strike, being a cantrip, has character-level based scaling, so that is kind of the ticket in terms or making up for the lack of 5th level class feature. So that’s +1d6.
So anyway… the Barb does two attacks at 2d6 + Str + 2 (assuming Rage) each. The Fighter/Bard does one attack only at 3d6 + Cha, but the 1s and 2s are considered 3s.
Assuming a +4 Str for the Barb (given their 4th level ASI), and a +3 Cha for the MC build, the averages come up to 13 damage per attack for the Barb, and 15 damage per attack for the MC.
Damage-wise, the Barb should come out ahead on average with the extra attack and reckless attack.
Defense-wise, I assume the Fighter/Bard probably gets better AC thanks to heavy armor and not having reckless attack cause advantage against them. If Bladeward is on, even more so.
HP-wise, the Barb should be ahead thanks to slightly bigger HDs and damage resistance, but the Fighter/Barb has plenty of Bonus Action healing for themselves (SW) or even others (HW), so durability should be quite good for both.
The big question is whether the Fighter spends their AS or not, and what for. AS is only slightly more scarce than Rage, since both recharge one use per short rest, so there’s only one more use of Rage per day than AS, given equal short rests (though of course Rage can last more than one round). AS can be used both offensively or defensively. Against a punishing opponent, it could be used to still attack while also bringing up Bladeward or Mirror Image via the regular Action. It could also be used to attack twice. Either way, the AS Attack action would yield an average damage of 11, since no True Strike. 13 x 2 = 15 + 11 so the average damage is tied in an AS round (assuming no reckless attack or that both somehow have advantage).
Anyway, it remains handwavy, and I’m not going to take a firm stand one way or another. But I’ll say, directionally, that it seems to me the Barb is stronger on offense and weaker on defense. How much either one matters in a given fight is case by case.
All in all, even though the scaling of True Strike is not as advantageous as the scaling of Extra Attack, it is not immediately clear to me that one is abysmal… maybe still weaker overall, but not 50% of the strength as one could assume thinking exclusively of Extra Attack. Maybe it’s more like 2/3 or 3/4 of the strength. Again though, very situational…
