When the neutron binds gravitationally to the neutron star, it loses a significant fraction of its mass.
Not immediately. When things falls they don't gain or lose energy. What really happens is more space is created above them and this makes them look (to people far away) a bit similar to falling. But the people far away don't think they are more massive. They think the star still has mass $M=C^3/(GT^22\pi)$ where $C$ is the circumference of the orbit of the people far away and $T$ is the period of the orbit of the people far away.
To understand this, first note what the word general in general relativity means. It doesn't mean that you can have acceleration (you can do that in special relativity). It means you deal with the general case when there aren't global inertial frames. All the physics you know is about inertial frames and there aren't global frames in general relativity.
So the frames themselves fall if you will. And if you have a shell of neutrons then as the star collapses the local frames fall, this creates more space outside the star which to far away people looks like the shell falling, but doesn't look like any local change at all to people far away, the spacetime out there does not change in the slightest. And the mass of the star does not change because of the collapsing shell.
If you imagine a funnel with a flat piece of disk shaped cardboard on the center then the people outside see a curved a spacetime, and the size and shape of the funnel is the mass they see.
When something falls what happens is that piece of cardboard is replaced with a smaller disk of cardboard. The spacetime farther out didn't change, the circumference didn't change. But now if you travelled in towards the cardboard and across it and back to the other side you would travel a larger diameter. So the circumference didn't change even though the diameter increased. That's because the funnel is curved and more curved space was created. (In reality, spacetime is curved so the star starts to also age more slowly relative to people outside.)
So the primary thing you think gravity does, the falling. Does. Not. Actually. Happen. Instead, space is created above you, your time slows down compared to other people above you and some more effects. But you don't fall.
What are these other effects? Remember how the cardboard disk was replaced with a smaller disk? So all the neutrons on the edge of that disk (the outer shell of the star) are now denser. The density of the shell increases. This is a tidal force, and is real and the neutrons in the shell move closer together and get more densely packed. In a realistic material this can increase the pressure and the temperature.
And that is key. The neutrons heat up from the increased density as the shell collapses and they radiate away that heat. When they radiate away that heat, the star becomes less massive. It becomes less massive because energy actually travels up in the form of light from the radiation from the heat.
If it slowly gets smaller it is because the temperature below has pressure keeping it large and as the temperature leaks out to the outside the pressure goes down and the shell collapses thus increasing the pressure again. And it is the energy that gets radiated away that allow it to get stuck as a smaller more compact object. The more and more that gets radiated away allows it to get smaller and smaller.
I would naively expect that the strong interaction would behave the same way regardless if we are at the surface of a neutron star or at the surface of the Earth.
Yes and no. It is fundamentally the same. Except all interactions for fermions are modified by a exchange interaction due to the superselection rule associated with the Pauli Exclusion Principle. This does act like more pressure when things are closer together.
And of course, more dense does also affect the pressure and temperature so whole of thermodynamics and the phase of matter and the equation of state all matter.
Inside the neutron star, the high pressure has an effect of putting the quarks closer together and decreasing the neutron mass, but this is a separate effect (we can see that if we consider the case where the neutron is just outside the surface of the neutron star as compared with a long distance away -- the closer the neutron, the lower the neutron mass solely due to the gravitational potential).
That isn't right. Pressure does have a gravitational effect (just like stress and momentum and energy). But increased pressure is like more mass. And your statement that the gravitational potential (not a real thing) lowers a neutron mass (it doesn't), is completely and totally wrong.
Firstly, the mass of a star isn't now and never was the sun of the masses of the parts and anyone that told you otherwise lied. Secondly as the shell collapses the absolutely only reason the mass of the star goes down at all, is because some energy is radiated upwards by light as this happens.
If some energy goes up in the form of light then the energy behind is less and that's why the star eventually seems less massive. Its not until that light gets to you that you see and feel a less massive star.
So how can the gravitational interaction influence the way the strong interaction generates the mass?
It doesn't. The mass of each neutron is the same. The mass of the star was never equal to the sum of the masses of the neutrons in the star. When the neutron star lets energy escape that allows the layers to contract, that contraction creates new space between where the outer layer used to be and where it is now. That new space and its creation is what the far away people think is falling. The people on the star see it getting more dense and they see some energy getting radiated away. The increased density is from a tidal force and the fact that the surface area of each layer of star gets smaller as it contracts. But each layer moves through the newly created space the layer below it made.
You destroy the space below you as you move into it and you create space above you and since you create move above you than you destroy below you it looks to far away people as if you are falling. But its just destruction a and creation of space and because of the geometry your density increases. The increase in the density is real. And the energy from this process of getting denser in the sense of having less surface area for your layer, that's real. And that energy can be given away. And if you aren't collapsing at the free fall rate then there is additional pressure from the layer below you doing work as you fail to move inertially and that can also be radiated away.
Your decreased mass of the star is 100% entirely due to the energy radiated away.