Earth's magnetosphere traps charged particles, electrons and protons, from the solar wind and from cosmic rays, in doughnut shaped zones around Earth. The higher concentrations of charged particles in the Van Allen belts means that there is a lower concentration of charged particles in the space between the two Van Allen belts, which is sometimes called "the safe zone".
Jupiter doesn't emit charged particles, it traps charged particles from the solar wind and cosmic rays within zones of its magnetosphere.
So it is possible for a moon of a Jupiter like gas giant to orbit within a gap between the radiation belts around the giant planet. Even if a moon doesn't have its own magnetic field, it can be protected from charged particles if it orbits within the empty levels of its planets magnetic field.
Here is a link to a question about empty regions in a giant planets magnetic field.
Are there areas within Jupiter's magnetosphere without powerful radiation?
Titan, the largest moon of Saturn, has an elliptical orbit, and from what I have read, spends part of the orbit within Saturn's magnetic field and part outside it. And I don't know whether Saturn's magnetic field hinders or helps Titan retain its atmosphere.
On the other hand, if a moon orbits within the radiation belts of its planet where charged particles are concentrated, it will need to have a magnetic field of its own to divert and trap the charged particles. And its magnetic field will have to be strong enough to handle the intensity of charged particles.
So it would be useful for you to find charts of the magnetic fields of Jupiter and Saturn showing the radiation belts and the areas of less intense radiation. Then you could select an orbital distance for your Earth sized moon that would put it in a safe zone of the magnetosphere or outside of the magnetosphere.
You wrote that you want your Earth sized moon at the orbital distance of Io around Jupiter, and:
You don't need to concern yourself with how this Earth mass moon would interact with Jupiter's other moons, or what the effects of Jupiter's tidal flexing on Earth would be (which I'm well aware would be MASSIVE and orders of magnitude stronger than the ones Io experiences due to Earth being larger).
But I can't ignore the effect of being that close to a Jupiter mass planet. The Tidal interactions would cause a lot of tidal heating of the moon's interior which would make its way to the surface.
If your planet orbited at the distance of Jupiter or Saturn, the tidal heating could in some cases provide enough heat to keep the moon at Earthlike temperatures, while the fraction of light received from the star might still be enough for photosynthesis.
But if the planet and moon are at 1 AU from from a star with the luminosity of the Sun, the combined heat from the star and tidal heating should heat up a moon in Io's orbit to not only produce a runaway greenhouse effect, but also turn it into a volcanic wasteland like Io.
The article "Exomoon Habitability Constrained by Illumination and Tidal Heating", by Rene Heller and Rory Barnes, 2013, considers this and other aspects of the hypothetical habitability of exomoons orbiting giant exoplanets. In this article Heller and Barnes coined the term "habitable edge" for the closest distance to the planet a moon could have and remain habitable.
In another article, "Magnetic Shielding of Exomoons Beyond the Circumplanetary Habitable Edge", 2013 Rene Heller and Jorge I. Zuluaga discuss the evolution of the magnetic fields of giant planets, since being inside the magnetic fields of their planets can affect the habitability of moons for better or for worse.
So I think that orbit you have selected, with the semi-major axis of Io, is much too close to a Jupiter mass planet, and would result in the moon having too much tidal heating for habitability.
So you should give your moon a wider orbit around the giant planet.
But you don't want to give your tidally locked moon a much wider orbit, because that would result in a much longer day night cycle and your are afraid the changes in temperature will be too much.
But a tidally locked moon of a giant planet, would have an orbital period, and thus a day night cycle of days, weeks, or maybe a month or two. That is almost infinitely shorter than the day night cycle of a planet tidally locked to its star, which will last for billions of years.
Of course, it is often believed that a planet tidally locked to its star, with one side in eternal light and the other side in eternal night, would be uninhabitable. The water and atmosphere might all freeze solid on the cold side.
But some studies indicate that tidally locked planets with the right atmospheres could have the right temperatures for life on both sides.
And if a planet with eternally long day night cycles might possibly have the right temperatures for life, a moon with a day night cycle a week or a month long could possibly have the right temperatures for life.
And it is possible for a moon orbiting a planet far farther than Io orbits Jupiter to have a day night cycle as short as that of Io, or maybe even shorter, if the planet is massive enough.
Planets can get much more massive than Jupiter without getting much larger in radius than Jupiter. As mass is added to a Jupiter mass planet its increase in size will slow down, and at a certain mass, stop. Adding mass beyond that point will actually shrink the size of the planet.
Planets can have up to about 13 times the mass of Jupiter and then transition into brown dwarfs. Brown dwarfs can have up to about 75 times the mass of Jupiter and then transition into very low mass stars. And I read somewhere that planets more massive that Jupiter have diameters within 15 percent larger or smaller than that of Jupiter.
So your planet can have an orbit several times as wide as Io's, beyond the habitable edge of its planet, while having a short day night cycle like that of Io, if it orbits a planet with several times the mass of Jupiter.