It's a great question!

Incomplete answer ending with a question:

Lagrange points are mathematical concepts that were derived assuming only two massive bodies (in the universe!) in circular orbits around their center of mass.

There will then be a gravitational potential field, and its gradient is a force field.

In the rotating frame those can be expressed as a pseudo-force field and pseudo-potential field. Looking at it in a rotating frame, the two massive bodies are fixed, and this makes solving some problems easier.

As you've mentioned already, the third body of the "circular restricted three body problem" or CR3BP is basically a "massless test particle" and you can put it anywhere to see what happens. The Lagrange points are mathematically derived places where a hypothetical test particle would sit still if placed exactly at the point. Just like a bowling ball would sit at the top of something. Depending on the shape, it might be stable against a tiny displacement (inside the caldera of a tall, inactive volcano), or it might be unstable (at the convex top of a pointy mountain).

The third body doesn't have to exist. It's an abstraction that we move around to answer "What if?"

...how big could that secondary moon be before it no longer fits the 'comparatively negligible mass' category?

There are of course more than two bodies in the universe pulling on each other, and even just in our solar system. And none of the orbits is exactly circular.

So we've already broken both assumptions necessary for Lagrange points to exist mathematically. If you put something at the Sun-Earth L1 point, gravitational effects from the Moon and Jupiter, and perturbing effects because the Earth's orbit is elliptical and not perfectly circular ruin any chances of putting something "exactly" at L1. It's not really there.

Instead there's sort of an "L1-ish area" where things will be somewhat more unstable than they would be in the pure CR3BP.

Adding mass to "the third" body in our actually many-body, elliptical orbit-based solar system won't make much of a difference to its trajectory until it starts changing the motion of the two bodies we're focusing on when pretending that the Lagrange points exist.

So if you removed Earth's Moon from its normal orbit and put it at Sun-Earth L1, That's about a 1% effect.

note: The best way to think of an object at the Sun-Earth Lagrange points is that they are orbiting the Sun just like the Earth, but in a resonant orbit with the Earth.

If the Sun and Earth orbited in circular orbits and there were no other planets, I think that they could still be in 1:1 resonant heliocentric orbits, but the distance will now be a little different than the classical L1 distance.

To double check on this, I've just asked

• Could the Moon placed near the Sun-Earth L1 point remain in a heliocentric 1:1 resonant orbit with Earth? If Earth were at 1 AU, where's the Moon?

in Astronomy SE, and will update here based on the outcome there.

It's a great question!

Incomplete answer ending with a question:

Lagrange points are mathematical concepts that were derived assuming only two massive bodies (in the universe!) in circular orbits around their center of mass.

There will then be a gravitational potential field, and its gradient is a force field.

In the rotating frame those can be expressed as a pseudo-force field and pseudo-potential field. Looking at it in a rotating frame, the two massive bodies are fixed, and this makes solving some problems easier.

As you've mentioned already, the third body of the "circular restricted three body problem" or CR3BP is basically a "massless test particle" and you can put it anywhere to see what happens. The Lagrange points are mathematically derived places where a hypothetical test particle would sit still if placed exactly at the point. Just like a bowling ball would sit at the top of something. Depending on the shape, it might be stable against a tiny displacement (inside the caldera of a tall, inactive volcano), or it might be unstable (at the convex top of a pointy mountain).

The third body doesn't have to exist. It's an abstraction that we move around to answer "What if?"

...how big could that secondary moon be before it no longer fits the 'comparatively negligible mass' category?

There are of course more than two bodies in the universe pulling on each other, and even just in our solar system. And none of the orbits is exactly circular.

So we've already broken both assumptions necessary for Lagrange points to exist mathematically. If you put something at the Sun-Earth L1 point, gravitational effects from the Moon and Jupiter, and perturbing effects because the Earth's orbit is elliptical and not perfectly circular ruin any chances of putting something "exactly" at L1. It's not really there.

Instead there's sort of an "L1-ish area" where things will be somewhat more unstable than they would be in the pure CR3BP.

Adding mass to "the third" body in our actually many-body, elliptical orbit-based solar system won't make much of a difference to its trajectory until it starts changing the motion of the two bodies we're focusing on when pretending that the Lagrange points exist.

So if you removed Earth's Moon from its normal orbit and put it at Sun-Earth L1, That's about a 1% effect.

note: The best way to think of an object at the Sun-Earth Lagrange points is that they are orbiting the Sun as the Earth is, (about 1% closer to the Sun), but in a resonant orbit with the Earth.

If the Sun and Earth orbited in circular orbits and there were no other planets, I think that they could still be in 1:1 resonant heliocentric orbits, but the distance will now be a little different than the classical L1 distance.

To double check on this, I've just asked

• Could the Moon placed near the Sun-Earth L1 point remain in a heliocentric 1:1 resonant orbit with Earth? If Earth were at 1 AU, where's the Moon?

in Astronomy SE, and will update here based on the outcome there.