What you have is a clash of expectations. Your group likes to plan and strategize, she clearly doesn't (or rather, planning and strategizing is less important to her than playing this impulsive character). It's really important to articulate the expectations of play and you should do talk to her, and the group, directly, rather than create an NPC to try to block her character.
You'll notice if you look at other media or stories, there's a big difference between impulsive characters who TALK about jumping in, but don't actually do so, vs. those who DO and cause a lot of chaos.
If your group expectations are that everyone is mostly working together and only conflicting/at odds for show ("The elf and the dwarf are arguing again." "Oh, let them be, they're actually good friends.") and she is operating on the idea that each character might have very different agendas or less concern for the group, you're going to have clashes.
Being clear about what kind of game this is, helps her decide if this is the game she wants to play or not.
1. Get a cheatsheet into each player's hands.
You know that godsend player, the one who always has the notecards? Key thing there: the notecards.
You've spoken to the group, and they got upset, but you know they cared enough to get the books in the first place. It's entirely possible that they do just forget, or maybe they're having a difficult time with the rules and don't know how to articulate it. Even if they read cover to cover (and maybe they already have), there's a lot of information to parse and remember, especially for new players. Even veterans forget rules sometimes! Still, fumbling through the book for basic rules is not time-effective. My solution is to create a cheatsheet. This has helped me as both GM and player.
Figure out your most commonly used rules. Type up these rules and print a copy for each player. Try to condense them as much as you can. Strip them to their bare bones so players can find the information they need at a glance. Also, since you'll be distributing the same page to everyone, concentrate on rules that everyone uses. I'll discuss character-specific rules later on.
It's just more practical for you to make the cheatsheet. If they don't have the rules down yet, they probably don't know which ones are the most important. You likely have a better grasp, and you're also the one who wants more efficiency. The best way to ensure that this happens is to do it yourself, and then the whole group will be on the same page.
For character-specific rules, though, it's best if the players do it. This spreads the effort around. Plus, you can use this as an opportunity to help re-familiarize your players with their mechanics. Sit down with them and help them make index cards or type notes from the book. Point out nifty features and answer any questions.
Assure them that it will make their lives way easier, because it will. They probably don't like book-fumbling either. They'll also be more likely to use their character's features if they're always right in front of them in an easily-digestible format. It's hard to achieve your full potential if you don't know where to find it. The key thing is to emphasize how useful it will be for them (in character and out) if they do this, not how much worse things will be if they don't. Make it about them and their potential fun, not your frustration, to avoid further souring.
It's awesome that the Battlemaster Fighter took the initiative, though. If this is something he enjoys doing, you could enlist his help! I'm this player in my own groups, even when I'm not GMing. I love having a cheat sheet, so if I'm making one anyway, why not share? At the least, use his notes as an example (if he's okay with it).
2. Make use of existing resources.
Cheatsheets you make will be the best-tailored to your needs. That doesn't mean you shouldn't use ready-made tools on top of that. People often share their own tools if they think they're especially useful. If you need something specific, search the web to see if it already exists. There's no guarantee that what you find will be useful -- again, you know your own needs best -- but it never hurts to try. If you hit gold, that's time and effort saved.
For instance, here are some form-fillable initiative cards that provide a quick reference of initiative order and PC skills, traits, and actions; here is a larger, longer version of a similar concept. Here's a site that generates spell cards. ENworld has many sheets available. It only took me a few minutes to locate them by searching "5e spell cards," "5e initiative cards," and "5e character sheets," respectively. You'll be more successful if you already know what you need, but you can still try searches like "5e game resources" and "5e player print outs" if you're looking for tools in general.
Best Answer
Be the voice for the voiceless
One of the lessons learned from years of "anti-harassment" training videos at work is that there are people that want to speak, but don't want to talk over other people (like me) and so rarely speak. It's also these people that when they do speak they have great contributions.
But the first thing you should do is talk to the soft voices outside of the game and see if they want help. Some players are happy just working from the sidelines. During combat they get their turn to shine and are done.
Assuming these players want to be heard more often, one method is, since you are capable of being one of the loud voices, to actively engage the softer voices.
If you use your voice to call on another, it may stop others from just shouting their opinion into the wild. You are making sure someone else has the floor, even if it's just for an audience of one; you.
If that still doesn't work, then reiterate their message.
If this still doesn't help, you'll likely have to involve the DM. Away from the table, talk to the DM and see if they are willing to control the conversations.
While a bit too formal for my tastes, I knew one DM that had an "out of combat" initiative. Basically at the start of each session, everyone rolled initiative (straight d20 as it's for the player, not the character). That would be the order the DM asked "What would you like to do?"
This way, everyone got a turn, and you weren't always right before or right after the same people. For intense conversations it would loop around to the beginning until the party had a unanimous decision.
It really boils down to what the soft players want.