The James Webb Space Telescope (JWST) is in a halo orbit around L2, at a sufficient radius around the Lagrange point that it is in perpetual sunlight. That allows it to have predictable solar power, but requires the very large and fragile five-layered heat shield.
What would the the design trade-offs have been for choosing a low-radius, perpetually shadowed L2 halo orbit and using a nuclear battery (such as a Pu-238 RTG, which is relatively easy to shield) for power?
That would be great, but the problem is that there aren't any orbits like that.
The only way to keep the temperature of the telescope rock-solid steady is to keep it in constant sunlight and insulate the heck out of it.
Going in and out of eclipse would cause all kinds of thermal perturbations, and the L2 Lagrange point itself is just a little bit too far away to be in Earth's umbra, and probably (though I don't know for sure) station-keeping so close to L2 would be a lot harder than doing it in JWST's big halo orbit around L2.
In no particular order:
What would the the design trade-offs have been for choosing a low-radius, perpetually shadowed L2 halo orbit and using a nuclear battery (such as a Pu-238 RTG, which is relatively easy to shield) for power?
Wikipedia says that JWST has about 2,000 watts of solar power.
Wikipedia says that NASA's flagship RTG, the Multi-mission radioisotope thermoelectric generator has an output in the beginning of about 2,000 watts of thermal power and only 125 watts electrical, after say 10 years that might be only 105 watts.
So you would need about twenty 45 kg RTGs, or another 900 kilograms to equal the power output of the much, much lighter solar panel, a far, far simpler technology you can almost order out of a (very fancy space) catalog these days.
JWST is receiving sunlight because it is relatively near to the sun. At Neptune, the amount of sunlight is much less.
As I understood, the heat shield is to shield the telescope from the heat of the sun. The telescope should cool down to just a few dozen kelvin in order to have low noise in the detector.
The solar panels are different from the heat shield. Solar panels are probably a much simpler source of energy than a nuclear option, and less dangerous in case the telescope would crash down to the earth.
Also, a nuclear power source would produce heat that needs to be shielded from the telescope that needs to be very cold for low noise observations.
Solar panels are black to absorb sunlight while a heat shield is reflective to absorb as little radiation as possible.
Solar panels unfolded 30 minutes after launch in order to communicate with ground: NASA site
The sunshield deployment starts at three days after launch, and ends at eight days after launch, in several steps, see: NASA deployment information
Please refer to nasa.gov for more details on James Webb.
As others pointed out, the L2 point is not in full shadow. Even if it were, the telescope would still need the heat shield to protect it from infrared radiation coming from the Earth and Moon.
Aside from the weight problem (a nuclear power source is a lot heaver than an equivalent solar panel in this part of the solar system), parking right at the L2 point would make it harder for the telescope to "hear" signals from Earth. The Sun produces a lot of radio noise and if it is right behind the Earth, then picking out radio signals from the Earth will be a lot harder. (A lot like trying to see a traffic light when the Sun is right behind that light.) By using a halo orbit, the satellite can point the high gain antenna at the Earth and the Sun's noise will be off to the side. (Similar to blocking the sun with your hand to better see the traffic light.)
This answer and the other answers to that question should help with understanding some of the drawbacks to parking right at the L2 point.
Your question suggests you haven't understood the importance of the nature of the L2 point.
Useful links to read more, if the brief description below isn't enough:
The L2 Lagrange point isn't an object. It’s a point in space where forces balance out to create a point that James Webb Space Telescope (JWST) will easily stay near. The "halo orbit" means it won't orbit exactly in an ellipse as usual, but in a slightly different varying curve, that needs correction now and then.
And that's the answer to your question.
.....at a sufficient radius around the Lagrange point that it is in perpetual sunlight. That allows it to have predictable solar power....
There isn't such a place as you visualise, there. L2 isn’t an object. Nowhere is "in the shadow", and you can't get into shadow by hugging it closer, because there's no solid object actually at L2, for JWST to orbit, like it would a planet or moon, and be in the shade of.
(Technically, L2 is one of five places where an object will broadly stay in the same position relative to Earth, with very minimal course correction being needed. Some matter may gather at or near L2, but no solid object in the sense you're thinking, able to provide shade and attract an orbiting satellite)
100 metres from L2 or half a million km from L2, it’s all the same thing - perpetual sunlight exposure (or more exactly it’s sliiiightly in Earth's shadow, so somewhat reduced sunlight) - but nothing actually there for it to hide behind.
