My read on Bianca is that she has an inflated ego from too many years of being the smartest person in the room. To be fair, she is a brilliant inventor, but it takes a special level of arrogance to claim you're ten times the smith as Branka, who forged a control rod centuries after the knowledge of how to do so was lost.
So anyways, she's asked by her old flame to figure out how red lyrium works. She ignores his warnings and travels to Bartrand's Folly. Examining red lyrium only yields so much information, but she can't leave the mystery alone. So she goes behind Varric's back to consult with a Grey Warden, who turns out to be Corypheus' vessel.
Red lyrium starts popping up all over and Bianca realized she unintentionally did a bad thing, but she doesn't want to admit she made a mistake. To ease her conscience, she tries to enlist her old beau and the Inquisition to help seal the thaig's entrance, hoping they won't put two and two together. (It's worth noting that this is risky to Varric personally, since the Merchant's Guild will kill him if they're together.)
After Varric and the Inquisitor prove that they're NOT drooling idiots, Varric gets a bit angry with her ignoring his warnings and hiding the truth. She doesn't respond well to guilt or criticism, so she first lashes out at Varric and then the Inquisitor.
Yeah, huge ego on this one.
To be just as fair, Orzammar is stuck in the past and so as brilliant as Branka was as a Smith, there's an entire wealth of information, technology and innovation on the surface that's being actively ignored by them. If Orzammar found out the means to make the repeating crossbow, would they use it knowing that it was created by surfacers? I doubt their stubborn pride would let them, even if it gave them an edge against the Darkspawn.
Sure, Bianca's boasting here and definitely does have a bit of pride in her work, but how is that any different from Dagna doing the same? Part of what made Dagna such as a great Arcanist was because she was able to learn new things on the surface that simply weren't part of Orzammar's dogged traditional thinking, so why shouldn't surfacer Smiths like Bianca and Gerav not be considered better than those in Orzammar who are stuck doing things old school?
She did screw up admittedly by going to Bartrand's Folly, but it was the only place to find Red Lyrium to study (at least prior to Inquisition), as either the shard was destroyed and Meredith's remains are being quarantined in the Gallows. If the shard was kept then Bianca did have a sample to study and designed the container to store it in, so it does come across as an even dumber move to venture into the thaig. Of course, studying the shard (if it exists) doesn't raise any clues as to what caused it and where it comes from, answers that can only be found perhaps at the source, which meant having to venture into the Thaig itself.
Contacting the Wardens for leads on Red Lyrium isn't that bad of an idea when you think about it, because if you're looking for information on weird stuff found in the Deep Roads, the Wardens are pretty much the experts in that field. In between games, Hawke does the same thing for those same reasons, leading them to team up with the Warden-ally by the time Inquisition rolls around.
I think she probably does have a prideful and stubborn streak (although she's a dwarf, it's kind of their thing), so she tends to rush into things thinking she's accounted for all the variables that could go wrong. Which of course leads her to overlook something as simple as human (or dwarven) error on her own part. I dunno, whether that's arrogance or a sign of her having flaws, probably depends on who's looking at it. To me, I think that kind of grounds her a bit and shows us that she's not above making mistakes and is just as fallible as anyone else.
It might be that part of the reason she was so defensive during that argument was because she didn't expect Varric to lay into her right there and then. Their conversation on the way through Valammar tells us that they'd not written to each other since before the Kirkwall explosion, so she doesn't know that he's assumed the mantle of all this blame on himself, as we see in numerous conversations during the game.
Varric does say that they'll probably reconnect at some point down the road eventually, but I personally don't expect it to be for a while, as it probably will take him a while to forgive her for this screw up.