Do you know that Base DMG, dlvl, attack power, and STR are all separate things, but combine to give you the DMG results?
Ifrit's Axe --- 171 Base DMG, +30 attack power
Barbarian's Bardiche --- 151 Base DMG, +7 attack power
The difference in it's base forms are obviously different being that Ifrit is greater, but let's take in account this Bardiche is triple melded.
If bardiche was melded with three IV tiered attack materia it could have a whopping total of +97 attack. The way that attributes work as of this last patch is in both modifiers only support attack. For instance, a MRD with a moderate amount of STR and VIT DMG modifiers floating around the 240-250 marks would only be suitable for mobs in a 45-48 Level range. Rather than someone who pushes against mob VIT or exceeds this limit on their attributes which in return pushes the attack through. Think of attributes as a latter trying to get over a VIT wall to combat mob's DEF. So, actually, yeah the numbers being +97 attack applied to one weapon looks awesome, but when it comes down to it, it'll show you incredibly inconsistent DOT.
You can mix up your materia and apply DMG attributes and attack and that would give you pretty substantial results. These builds greatly vary though between fighting Ifrit and Fighting moogle. Moogles have a VIT and DEF rating of the class they are in the Lv50-53 range. So, in order to hike that DOT way up you need your modifiers both at 280+. With Ifrit, this is a completely different story as his VIT is very high. So, this weapon is completely a head ache already since you need to develop it to be situational.
Now we'll get to base dmg. Your biggest problem in your declaration to a triple melded bardiche being better than and ifrit's battleaxe. Attributes and attack have soft caps, but they are based on your base DMG.
Let's visualize a staircase that has exactly ten steps. In Bardiche's case, it only has the power to climb at least to step 3 and max to step 5 before it tires out. Ifrit's Battleaxe has the power to climb to at least step 4 and max step 6 before tiring out.
The base DMG alone is enough to tell you that the weapon is superior even with the bardiche's potential triple meld. Don't get me wrong, a triple melded bardiche is incredible for what it is, but a high base DMG weapon + a moderate amount of attack power is always going to out parse it. End of story. Provided that the player knows what he/she is gearing for.