Hawke is very good in killing, essentially a killing machine that is true. But is also a lazy idiot which disqualifies him from the title of badass.
"Want to cause a war with the Qunari Petrice? Eh I cbf stopping you"
"Want to take that list of Qunari Spies so nobody can find out who they are tallis? Eh I cbf stopping you"
"Want to take my sister to the circle despite my clear dislike of this and my family trying to avoid exactly that for the past 18 years? Eh I cbf stopping you"
"Meredith taking all power in Kirkwall and blocking the new election of A Viscount so she can put a puppet on the throne? Eh I may say something about that in 3 years, but cbf doing anything other than saying a few words"
"Guy just walked past me with the tome of Koslun, The very reason the Qunari are here and the sole thing we came to this warehouse for? Eh I cbf going after him, I have Tevinters and Qunari to kill while they're killing each other"
No badass here
Ignoring all the factors that prevented Hawke from taking action against these people, you mean?
Petrice was a respected member of the Chantry, while Hawke was a penniless refugee with absolutely no proof she'd tried to incite a race war with the Qunari save their word against her, something Petrice even lampshades. In Act 2, while both have increased in position, Hawke still has absolutely zero concrete evidence against her save that which is circumstancial, something that you're unfortunately forced to admit when the Viscount asks if you have any proof of her involvement?
While we're railroaded to not try to take the list from Tallis, she does mention that the list doesn't just contain the names of current Qunari spies, but those who've since renounced the Qun and become Tal-Vashoth, many of whom might have established friends, family or loved ones as part of their cover who would be in the line of fire if the list was exposed. Do you really think the justice system (if it can even be called that) in Thedas would recognise some people might not be aware of their friend/family/boss' affiliation with the Qunari? Bull's name being on that list and being outed would probably mean that the Chargers would all be either imprisoned or executed as accomplices in his espionage, regardless of how complicit they were. Even a completely loathsome Hawke who'd sell Fenris into slavery, murder Merrill's clan or betray Isabela to the Qunari, probably isn't enough a complete tool to let thousands of innocent people die because they happen to know someone in the Qun.
Hawke doesn't stop the Templars taking Bethany because there is nothing they can do... Bethany had already been outed and save for attempting to flee the city, something none of them want to do, all of them are now at risk because they've been knowingly harbouring an apostate for years, a serious crime in of itself (especially as far as Meredith is concerned). That Bethany chose to go willingly (and might have been the one to out herself), as well as Hawke not attempting to prevent the Templars taking her is part of the reason that Cullen probably was able to wrangle some leniancy for them. Furthermore, as much as Hawke doesn't want Bethany to go to the Circle, it was her choice to go.
Aside from being a minor member of the nobility, Hawke has no real power in the city whatsoever (something Aveline mentions in Act 2, that Hawke has no job) and even becoming the Champion makes them more of a mascot for Kirkwall, than holding any kind of office. Meredith was already running the city unofficially when Hawke arrived in Kirkwall and we see that she further consolidated power after the death of the Viscount. In Act 3, Bran and Aveline are the only two legitimate government officials left and it's made clear that Meredith was attempting to undermine Aveline to try to seize control of the guard, so the city would be under full Templar control. Hawke cannot act against her either, due to Meredith making veiled threats against either them (if a Mage), Bethany (if in the Circle) or Hawke's mage friends, so it's clear she's holding all the cards.
As for the entire situation with the Tome of Koslun, when all hell breaks loose and you're stuck in a melee a trois with Tevinter and Qunari forces, is it any surprise that maybe you might have more pressing matters to deal with (such as not dying) while the guy holding the book slips passed everyone, especially when Isabela bolts after them in all the confusion, further complicating matters?
Regardless of how badass someone is, some things are simply too complicated for them to take action against, as it'd either end in death, incarceration or suffering reprisals either against themselves, their family or their loved ones? One cannot live by murder-knife alone.
