Bard 4/Barbarian 1... -7 hp compared to a 5th level fighter, but can cast 2nd level spells with no ASF, and you get some niceties such as Hideous Laughter, Glitterdust, Invisibility, Heroism, Cure Light/Moderate Wounds, Alter Self, Blur... you also get a 40' movement (helps with the flight too) and rage 1/day from the 1 barb level. 4 BAB for 5 levels.
Or Bard 1/Barbarian 4.. more hp, only cantrips. Honestly I don't think the tradeoff is worth it
Bard 1/Fighter 4... less mobility, gets weapon spec which is overall better than rage probably.
I tend to like the first option.
A bard 4/barb 1/DD 10 has 11 BAB at 15th level, which isn't great, but when you factor in the +8 str you actually have the same to hit and number of attacks as a 15th fighter (ignoring greater weapon focus). You do get that iterative attack later, but you get natural attacks to make up for it.
Base saves are 10, 7, 11 as compare to a 15 fighter's 9, 5, 5.
HP are 6 + 3x3.5 + 6.5 + 10x 6.5 = 88 average HP (then +15 for +2 con) compared to our fighter's 10 + 14x5.5 = 87 average hp.
Then there's the DD special abilities.
Ok so 6 feats to make up the difference. 1 goes into improved toughness, then the 2 chars are at basically the same HP. It takes 2 feats to make up for the +4 damage/hit the DD gets (weapon spec + GWS). Thats assuming 1handers. For 2handers, the fighter can't entirely make up the difference. It would take 5 feats to make up the differences in saves, assuming Iron Will etc all stacked, which they don't. Then blindsense, flight etc you can't really take feats for. So overall, I'd give this build a big thumbs up.. it looks better than an equivalent level fighter to me.