If you want to go with the old school.Rawlsian position, Jonathan Quong would be useful for you. But I think that position has been pretty roundly rejected for its antidemocratic implications. Two other things: "deliberative democracy" is an umbrella term that basically encompasses all democratic theory these days. It just means that deliberation is at the heart of political legitimacy but in lots of different forms (so this includes Rawls and political liberalism). Also I wouldnt be so quickly dismissive.of epistemic democracy. The first citation you gave is from 2007 and epistemic democracy theory has really exploded since then. Some of what you sound like you're arguing is in fact for epistocracy or rule by experts, but if you read Helene landemore, she utilizes the "diversity Trump's ability theorem" which claims that a plurality of thinkers are better at solving problems than a small group of experts. The diversity approach also helps solve the problem of who counts as an expert since in a democracy we all have to debate who the expert is anyway.

So this sounds like Rawl's argument from Political Liberalism, but even Rawls changed his mind, see "Public Reason Revisited." Excluding personal or religious reasons from public deliberation is just a way of imposing a set of moral values on a democratic public without/before the process of democracy. Wolterstorff talks about this in his book with Robert Audi. A really good take down of this point of view is Christopher Eberle, Religious Conviction in liberal politics. Check out the sep article on public reason and all of the criticisms. Even the arch rationalist Habermas changed his mind about religious reasons in the public square.

I think your best bet is to go with including everybody and all their reasons in the most robust democratic deliberation possible. Go with an epistemic defense of democracy that argues the legitimacy of democracy comes from its epistemic ability to identify and solve social problems. Check out Helene Landemore and David Estlund. Epistemic democracy is a super hot topic right now and I think it makes the best argument for Democratic legitimacy.

I think you should read Fabienne Peters work. Her book Democratic Legitimacy is a good place to start. She also wrote the sep article on political legitimacy. This will help you categorize these various views you want to engage (normative v descriptive legitimacy, procedural v. Instrumental legitimacy, moral v epistemic approaches).

One challenge you're going to have is that if democracy is entirely instrumental then who is deciding these outcomes the democracy is meant to achieve? The point of democracy is that we all get together and debate what the good life or good society is without one person or group imposing their view on another without sufficient justification.

A second area to work on is defining what you mean by reason. One of the big debates within this field is what counts as a public reason and what limitations can be placed on reasoning. Kevin Vallier and G Gaus have written extensively on this. You argue that reason has legitimacy but not democracy, yet democracy is the arena where we share and debate reasons. If we had a God's eye view or AI computer that knew all the correct answers then we wouldn't need to share reasons but since we don't have that, democracy is the opportunity for us to deliberate as equals without imposing our own conception of whats easonable on others.