The list of questions are quite broad, but I can provide some starting ideas.
In terms of question 2, think about the available sources of media – it would be mostly newspapers at that point, which can very easily shape public opinion considering even factory workers would have news read to them (look up the historical lector). However, there is a disconnect between the elite/politicians and the common people. Despite this, significant riots and movements will grab the attention of the upper-class for certain.
Changing public opinion is not necessary to change the opinion of the aristocracy or politicians. You just need to provide enough incentive or threat. Given that a significant amount of politicians are probably (and this is a probably, not a guarantee, if the politicians are already against the breweries this becomes much easier for the Band of Hope) entrenched or linked with the breweries, persuasion through incentives becomes hard. How can you provide them with more money or power than in the status quo? Threats can range from economic to physical, you can get creative here. Who knows, maybe one of the politicians has family in a nearby city that operates a brewery whose competition is being shut out by the brewery in this city, and all it'll take to push him/her over is a small nudge or visit from the Band of Hope.
In terms of threat progression, there has to be something to rally the cause behind – in this case, religion/religious establishment would be one way to take it. Past marches can escalate into riots and destruction of factories/property with public support. Threats against key figures can always be effective. The terminal goal is probably Prohibition, and honestly this could happen at the rally/march stage with enough political support. Ultimately, this comes down to how willing the political elements are to the idea of prohibition, and how far the Band of Hope can push people to pressure political elements. This can be artificially manipulated both ways (on the political side, or inflating the seeming number of supporters in the movement/damage done).
The crucial thing to note here is that the Band of Hope needs to be able to somehow appeal to the average person. Why would the common worker who slaves for hours a day and relaxes with a beer want to shutdown a brewery? The motivation for this doesn't seem to be grounded in health concerns, given your description of the Band of Hope. Returning to a more conservative moral stance can work, but what would make it more successful than in real life? Either there is some anecdote to manufacture (or coincidentally happen), or there is some sort of slogan/key phrase, or a shift in prevailing attitudes, or some combination in order to really attract the common people.
Mitigation of this can be done through multiple means. Discrediting the Band of Hope through targeting key members or the general message are both possible. They are literally making beer, which they can leverage to sway public opinion. The owners of breweries probably also have significant sway in the political realm through their economic clout, they should be able to do whatever the Band of Hope can, make counteroffers etc. A very large part of the reason why Prohibition didn't work out was simply that there was too much public demand for it, fuelling the underground market. The brewers can easily appeal to the common people and their tastes to reverse public sentiment.
With respect to your first question, I think it's definitely plausible to the average person, if not outright plausible. It's an interesting clash you have set up here, and there are a lot of possibilities!