This propulsion system won't work like traditional systems that rely on a reaction mass and, therefore, conservation of momentum. Instead it will work more like reverse osmosis.

Reverse osmosis devices are used to make fresh water. They also conserve momentum. 

Conservation of momentum isn't just how conventional rockets work. It's a law that we suspect applies universally and without exception. 

In the reverse osmosis analogy, the "solution" is space-time and the "solute" is the vacuum fluctuations. In osmosis, the solution will try to equalize the concentration of solute.

Wordy analogy based reasoning of this kind does not reliably produce correct answers. 

At best, reasoning like this can be used to generate a suggestion for what equations to consider. Because if the upside is a nobel prize, and the downside is wasting a few hours, it's worth a go even if it's probably wrong. 

This relies on a person that understands the maths of quantum field theory. 

In practice, there are a lot of people going "I have the ideas, I just need someone to add the maths", and not that many people who understand the maths. 

Looking at your equations, I think I can spot at least 1 mistake. 

You say $F=T\frac{dS}{dx}$ from "Verlinde’s entropic gravity".  Verlinde's work isn't something I am familiar with, so I can't say whether this is correct or not. 

But, if it is, this is the force at 1 point. To calculate the overall force, we must take the integral. 

If $S$ tends to a constant (the background entropy of empty space) sufficiently fast as the distance from your spacecraft increases, then we can use the gradient theorem https://en.wikipedia.org/wiki/Gradient_theorem to show that all the forces must inevitably cancel out. 

Consider this diagram. 

 The entropy needs to be the same at the far left and far right of the graph, because empty spacetime, far from any influence, has a fixed entropy. Your spacecraft sits in the middle. A small amount of your spacecraft on the far left experiences a steep gradient, and so a strong rightwards force. A larger amount of your spacecraft in the middle and right experiences a weaker leftward force. 

And so it all adds up to 0 force in total. 

This sort of everything cancelling out behavior is an inevitability assuming the equation  $F=T\frac{dS}{dx}$ and

1. Flat spacetime
2. T and S are converge to a constant (and do so at least cubically fast, 2x the distance means at most 1/8 th as much variation in T and S.) 

So looking at the theory more in detail, I am not convinced this works very well. Generally speaking, Verlinde's theory doesn't seem very well accepted and is taken more as speculative (and Verlinde's own paper from 2016 about emergent gravity seemed to shift the focus a bit away from just entropy), but also, as far as I can tell, the problem is what is the main entropy contribution from a given region of space. And in Verlinde's derivation, he makes that to depend on all information that describes entirely that region. That's what the holographic principle means after all. Really basically his idea seems to be that, given a certain amount of matter, the configuration where those bodies are closer is higher entropy than one where they aren't; so the universe tends to bring them together.

The problem is that as far as I can tell there's no clear idea of how that would work out in more complex configurations of matter, and in fact that seems to be one of the main criticisms of Verlinde's theory; and if it's just an explanation of gravity, it shouldn't allow repulsive forces, which would instead be necessary for any kind of engine to work (otherwise you're not flying... you're just falling with style).

Also the scales involved will be dominated by $M{c}^2$ terms (all the energy of the rest mass of the configuration of matter which is distributed among the degrees of freedom encoding it on the holographic boundary), so honestly odds are anything you can do with the Casimir effect is trivial by comparison.