I'm currently working on a project with an inhabited terrestrial planet slightly smaller than Earth, and orbiting it is a moon, which is a similar size to that of Earth's moon. While our own moon has a small, cooled core, would it be possible for the said moon to retain an active core and maintain a magnetic field strong enough to retain an atmosphere?
Here is a frame challenge.
Having a magnetosphere to protect the atmosphere from being eroded by the stellar wind is only important if the world is massive enough to retain an atmosphere for long.
You should check Habitable Planets for Man, Stephen H. Dole, 1964 pages 34 & 35.
According to Dole the main determination on how fast a planet loses atmosphere into space is the ratio between the planet's escape velocity (the escape velocity, and not the surface gravity) and the root-mean-square velocity (which depends of on the exosphere temperatures) of atmospheric gases in the exosphere, the outer atmospheric layer where gases are lost from. Table 5 on page 35 gives the time it takes for an atmosphere to be reduced to 1/e, or 0.368 of the original amount, for various ratios.
Dole discusses the mass range for human habitable planets on pages 53 to 58.
Various other process can speed up atmospheric loss rates, but nothing can slow down the loss due to the world having insufficient escape velocity. The only way for a world with insufficient escape velocity to maintain a steady atmosphere is to produce or acquire gases as fast as they are being lost. Which is extremely improbable if the world has a ratio of escape velocity divided by root-mean-square of exosphere gas velocity which is low.
The Moon has an escape velocity of 2.38 kilometers per second. Earth and Venus, with dense atmospheres, have escape velocities of 11.186 and 10.36 kilometers per second. Mars and Mercury, with thin atmospheres, have escape velocities of 5.03 and 4.25 kilometers per second. Ganymede and Callisto, airless by most definitions, have escape velocities of 2.741 and 2.440 kilometers per second.
It is true that Titan, with an escape velocity of 2.639 kilometers per second, has a dense atmosphere. But Titan is very cold, including in its exosphere, lowering the gas velocities in the exosphere. If you want your moon to be so cold that the lifeforms drink liquid methane like hypothetical Titanian lifeforms would, then you can have a dense atmosphere on a world like the Moon - but of course the planet the Moon orbited would also be very, very cold.
The Moon, and thus any world similar to it, has far too low an escapee velocity to maintain an atmosphere for geological eras of time if it has temperatures similar to that of Earth.
One way a small world could keep it's atmosphere if there is a roof over the world to hold atmosphere in.
A planet or a planetoid turned into series of concentric matryoshka doll-like layers supported by massive pillars. A shellworld of this type features prominently in Iain M. Banks' novel Matter.
A megastructure consisting of multiple layers of shells suspended above each other by orbital rings supported by hypothetical mass stream technology. This type of shellworld can be theoretically suspended above any type of stellar body, including planets, gas giants, stars and black holes. The most massive type of shellworld could be built around supermassive black holes at the center of galaxies.
An inflated canopy holding high pressure air around an otherwise airless world to create a breathable atmosphere. The pressure of the contained air supports the weight of the shell.
Completely hollow shell worlds can also be created on a planetary or larger scale by contained gas alone, also called bubbleworlds or gravitational balloons, as long as the outward pressure from the contained gas balances the gravitational contraction of the entire structure, resulting in no net force on the shell. The scale is limited only by the mass of gas enclosed; the shell can be made of any mundane material. The shell can have an additional atmosphere on the outside.
The third type of shellworld would be the easiest way to put a roof over a moon.
The Moon once had a significant atmosphere.
In October 2017, NASA scientists at the Marshall Space Flight Center and the Lunar and Planetary Institute in Houston announced their finding, based on studies of Moon magma samples retrieved by the Apollo missions, that the Moon had once possessed a relatively thick atmosphere for a period of 70 million years between 3 and 4 billion years ago. This atmosphere, sourced from gases ejected from lunar volcanic eruptions, was twice the thickness of that of present-day Mars. It has been theorized, in fact, that this ancient atmosphere could have supported life, though no evidence of life has been found. The ancient lunar atmosphere was eventually stripped away by solar winds and dissipated into space.
If the lunar atmosphere would be lost into space in about a few thousand years, it would have had to be totally replaced every few thousand years over the period of about 70 million years that it lasted. Thus it would have been replaced tens of thousands of times over the 70 million year period. If the Moon could have retained all of the gases it emitted, it's atmosphere would eventually have been a few hundred times as dense as Earth's.
And of course the Earth had a dense atmosphere for billions of years before lifeforms eventually produced an amount of oxygen in the atmosphere sufficient for breathing and the Earth became habitable for humans and for other lifeforms which breath oxygen. So a moonlike world with a temporary atmosphere lasting only 70 million years would be unlikely to naturally produce a breathable atmosphere in so short a time span.
But if an advanced civilization built a roof over a moon like world when it was producing gases they could save the gases, accumulate a dense atmosphere, and introduce plants to produce oxygen.
The easiest answer is: Yes.
Just state that this moon is much younger than the Moon and that it was formed with a larger core.
The younger age gives less time for the interior of the moon to cool and stop spinning, and the larger core gives the likelihood of a stronger magnetic field.
Depending on how little scientific rigor is required, you could just state that both planet and moon have an atmosphere, and leave it at that without needing to go into details.
Fill it with enough fissiles, and anything is possible.
Fissile materials are a major source of "new heat" inside the earth (or at least we believe that's how it works) the percentage is pretty low, but if you had more of them to start with, and the right percentage had the right half life, it could stay warm inside for a very long time.
Tidal forces can heat things up too, but careful with that one! You don't want to end up with Io!