They're vaguely market anarchists. I've sort of imagined them more along Hoppean-Rothbardian lines (these are modern philosophers who were "anarcho-liberals"), but the historical precedents would be people like Gustave de Molinari, Max Stirner, Jakob Mauvillon, and Benjamin Tucker.
Essentially, these people believe that the modern, liberal state is bad, for whatever reason. Either because it violates property rights (taxes are theft, and anlibs believe property is the primary ethical principle, so taxes and the state are evil), because it grants bad types of property (the state grants titles to property that are not justified - these are the mutualists like Tucker who are actually very pro-market, only they think this system of distributing claims to property is bad), or for another reason (democracy means everyone plunders his neighbor, traditional clan-based aristocracy and crap is better, etc. etc.). Exactly what their objection to the modern state is and what their alternative proposed isn't really clear - what is clear is that they don't like the gob'mint, and they do like the free market.
I've always imagined that the bourgeois dictatorship is either:
A) A transition into this anarchist society - the anlib revolution has toppled the state, so the dictatorship is essentially a coalition of revolutionary vanguards that is temporarily exercising force (being illegitimate by collecting taxes, etc.) as a "radical, revolutionary government". So, the soviet Union was not a true proletarian dictatorship (becasue hierarchy remained in the state, etc.), just as anarcholiberal France isn't really anarcholiberal (because hierarchy remains in some Robespierre-type Reign of Libertarian Terror). I've always kind of imagined this being essentially a Committee for Public Safety situation, where the anlibs are going about killing former government officials (as retribution), putting together revolutionary purges, etc... but actually we've just ended up with a slightly more pro-market government.
OR
B) This really is an anarcholiberal society, and the bourgeois dictatorship is just the best analog through which the player can "represent" that society. Anarcholiberalism rejects the state, so there'd be no single, monolithic actor for you to play if we simulated this accurately. Law would be produced through multiple, overlapping courts and legal systems, there would be multiple private security organizations (which would probably be localized to protect their clients... so the military would be really complicated), and of course no taxes exist, etc. There's no government that decides who resources are allocated, so I've always just assumed that this "bourgeois dictatorship" you play as is essentially just the market itself - you're the invisible hand of the market. "Taxes" are just the total amount of voluntary consumer spending allocated to the vital sectors that the state would otherwise control, and the military consists of many coalitions of private organizations.
Just to be clear, "anarcholiberalism" (which today is generally known as market anarchism or anarchocapitalism - there exist nuances in these ideologies but they are both similar to the anarcholiberals in game) is a libertarian ideology, but not all libertarians are anarcholiberals. Libertarianism generally just means an ideology that places a value on human liberty, which can basically mean anything (and historically, geographically, and intellectually means hugely different things... this is why many libertarian socialists call libertarian capitalists non-libertarian. Chomsky does not think Ron Paul is a libertarian, and left-wing anarchists don't think that right-wing anarchists are really anarchists, and right-wing anarchists don't think left wing anarchists are really anarchists: Tucker hated Marx, and Chomsky disagrees with Rothbard). Anarcholiberals value private property and individual autonomy, so they are "libertarian", but most libertarians don't believe in abolishing the state to be replaced with private legal institutions.
Now, the reason anarcholiberals want to roll back political reforms is that democracy doesn't jive well with anarcholiberalism as an ideology. A system in which the majority of the population is entitled to (as anarcholiberals believe) expropriate property from the minority is really horrifying to anarcholiberals, who think property is sacred and the individual should be able to do whatever he wants with himself. If force has to be exercised, it should be exercised in defense of property, and the sort of institutions that would do that in an anarcholiberal society would probably not be democratic, at least in the sense we're thinking of. Anarcholiberals think the democratic, modern, liberal state is bad, and that state reflects a value system (populism/collectivism) that is very much opposed to anarcholiberal ideology.
Also, "19th century classical liberals" would generally just mean the liberal party. Like, the really radical wing of the liberal party would include the minarchists like Bastiat, but Bastiat and friends still would not find themselves in the anarcholiberal category. The anarcholiberal category is for people who offer a radical critique of the modern state and want to abolish that state with a market, so like Molinari and Mauvillon (and then... idk, Stirner and Tucker are difficult to categorize. I'd put them here despite their socialist sympathies, because they also really opposed the socialist ideologies of their time and meant something different than the modern sense of "socialism").
tl;dr anarcholiberals have an historical basis, they were just never a mass-ideology. It's basically a neologism that describes some very radical 19th century liberals and anarchists who modern day anarchocapitalists like a lot. They don't allow voting b/c there's no government, and they're both anarchists and libertarians.