Answer: If JWST goes “over the hill” past SEL2, it will enter a horseshoe orbit. But it will be non-operational in this orbit. A “rescue burn” of its thruster might restore its halo orbit.
As you said in the OP, JWST is currently in an unstable halo orbit around SEL2. If allowed to free-fall, it would likely “fall” past the Earth and follow a horseshoe orbit around the sun, alternating between being inside and outside the earth’s orbit. If JWST was mistakenly allowed to drift antisunward past SEL2, it would enter the same horseshoe orbit, but would start its first revolution around the sun beyond Earth’s orbit, instead of inside it.
I know that sounds weird. How can JWST end up in the same orbit if it goes in the opposite direction? Let’s look at horseshoe orbits.
This diagram is from Invariant Manifolds, the Three-body Problem and Space Mission Design G Gomez et al 2001.
To confuse you, the Sun is labeled “J” and the Earth is labeled “M”. To further confuse you, “Stable” and “Unstable” are not used in the usual sense. “Stable” trajectories are those which are heading towards a libration point, like L2. “Unstable” means trajectories which are heading away from a libration point. “Stable” orbits are drawn in green and “Unstable” orbits in red. After an object has been on an “Unstable” trajectory (away from a libration point) it will make a transition to head towards a libration point (the same or perhaps a different libration point from the one it just left). Its orbit now becomes “Stable”. These transitions from red to green and back are shown in the diagram with black transition lines labeled U1 –U4.
An object in a horseshoe orbit follows the invariant manifold. On one leg of the orbit, it is Sunward of Earth’s orbit, catching up to Earth. On the next leg, it is outside Earth’s orbit, falling behind.
An “invariant manifold” is a surface comprising all possible green/red orbits with a given energy level. So an object on any of the orbits which make up an invariant manifold does not need any delta-v during its orbit.
There are an infinite number of similar invariant manifolds nested inside each other, like telescope tubes.
Although it is difficult to appreciate from the diagram, “Stable” orbits become closer to their previous pass as they approach L2, forming a halo orbit.
Representations of Invariant Manifolds for Applications in Three-Body Systems / The Journal of the Astronautical Sciences, Vol. 54, No. 1,
These “Stable” halo orbits are almost identical trajectories to the halo orbits of the corresponding “Unstable” orbits.
JWST was launched on a “Stable” orbit towards SEL2. At insertion into its permanent orbit, it was left just shy of the L2 “Stable” halo in an “Unstable” halo which would return it towards Earth without occasional corrections.
The OP rightly pointed out JWST only has thrusters on the sunshield side. Since JWST is usually oriented with the sunshield side towards the sun (duh!), the thrusters are appropriately oriented to counter the Earthward drift of JWST’s “Unstable” halo orbit.
However, if a mishap pushed JWST beyond L2, there is no reason JWST could not change attitude for a corrective burn in the antisunward direction. The science instruments need deep cryo to function, but they were not damaged by years at room temperature before launch. As long as the image of the Sun, Earth dayside or Lunar dayside are not focused on the secondary mirror during attitude changes, instruments would not warm above room temperature and should operate normally once cooled back down.
This “rescue burn” would have risks, but the alternative would be functional loss of JWST.