lays out a view of the future where only the military is allowed full citizenship
Citizenship is not about military. It is tied to responsibility. As it should. And opposite as we have it most places in today's world, especially where "democracy" reigns.
And responsibility must be demonstrated up-start, and lasting as long as one's life.
In the ST world the setup has an ongoing war with serious enemy without chance for reasoning, so obviously military service is sensible way to demonstrate that one is ready to lay his life for fellows anytime.
I'm sure if you find a different way to demonstrate the same you're welcome as citizen and allow to rule over others.
He presents democracy as intrinsically flawed
What anyone with little mind can see, but H actually shells out pretty good reasons to back up the positions. One that anyone would welcome to address, just it's not happening. Both past and recent history shows the truth about it.
and physical punishment as an essential part of child-rearing.
The idea "Spare the stick and spoil the child" is present in many of his books. With all good sense too, in the original contexts carefully balanced with all kinds of factors. And explanation why alternatives suck.
The novel is often seen as advocating militarism if not fascism.
Yeah, but mostly those who can't read, who did not read, and essentially those who didn't reach even the first level in thinking. (And I'd be more than interested in conversation with whoever having that position and outside this set.)
By contrast, the 1961 Sci-Fi classic Stranger in a Strange Land seems to advocate the polar opposite view, and is partly anti-violence and anti-fascism.
Interesting, I never saw any contradiction between ideas in those books. And quite sure there is none.
It certainly can be created by unfair accusations and mincing words.
Let's see violence: is there any self-serving violence in ST? Show it if you can. And in Stranger Mike can just shift someone to another dimension, that is not considered violent? Well, violence is not the point, we just want the world in certain state.
Martian world just works differently, selection happens early and unattended. No one bothers with the larvae, using the stick or whatever. Majority is just left to die out naturally, and the few that survive are considered good enough as sentient being. And in a sentient society there is little violence. (See same idea in Moon and other books.)
And fascism is completely made up (mostly seem by people who you could ask to define or explain what the term means and see them baffled.)
How did Heinlein reconcile
With some hope he did not and should not. My idea of Heinlein that at some point he recognized same as Jubal -- people reject whatever saving efforts, and it's better if he just sticks what he thinks is fun. Those who approach the book with open mind and heart can understand them with ease. Most ideas are not really complicated, just become look like that due to massive amount of dirt added in our world. Once you learn to drop the facade and irrelevantia, can look just the original form, you can manage alone.