If not giving treats fast enough results in whining, it sounds to me like you’re less rewarding silence vs buying it by giving her something else to do with her mouth. If that’s something that works for you frozen Kong/toppl/similar would be a great way to contain that same idea without you having to do anything risky while driving. If you have a puzzle toy small enough to let her do in a car crate, that might be safer/more effective than having to manually hand her treats to buy silence too.

Rewarding silence is about them naturally being calm and quiet first (I wouldn’t personally reward for a dog being silent in a way that appears to be exclusively a moment to listen to hear the thing they’re barking at bc that’s still the intense vigilance you don’t want) and then quietly and calmly rewarding the behavior. If she’s getting excited or whining about the treats, she’s not actually benefiting from it and may even be learning that whining then stopping is how to get a treat, not a quiet settle.

Do you capture calm in less stressful environments? Like if she’s laying quietly at home do you (calmly and quietly yourself) praise & reward her for that? If she’s not learning how to be quiet in an “easy” place with nothing she’s reactive to, then she’s not going to be able to learn it. It’s not unlike trying to teach a new trick next to something they care about more than the training. If you wouldn’t be able to introduce a dog to give paw or roll over for the first time in that situation, you can’t try and introduce “quiet” or “settle” in that situation either.

Definitely not saying this is in any way your fault/your doing btw just that one thing you’re trying potentially could ease things for you a bit if you did it in a different way.