As you orbit the gas giant, it will appear to change it’s phase, waxing and waning, over one lunar orbit (let’s call it a “month”), just like Earth’s moon appears to.
Let’s imagine your first observation is of a waxing half-planet. During the observation you noticed that the ANGLE of the shadow-edge (terminator) across the moon, matched exactly the vertical line of the noonday sun.
After one month however, the angle of the sun, relative to that of the planet will have changed, due to the planet’s solar orbit inclination. Since you told us that on the moon, the noonday sun angle will not change throughout the year (a very peculiar, unnatural arrangement- that proves extremely useful), we can use it as a reference to measure the angle of the NEXT waxing half-planet terminator.
Here is an animation of earth doing exactly this over the course of 12 months:
If this were a circular orbit, the detected change in angle will allow one to compute the orbital inclination of the planet, assuming one knew how many “months” are in a “year”(planetary orbit).
Computing the eccentricity of the orbital ellipse requires the above information PLUS soe additional observations. The AMOUNT of angular change detected each month in the planet’s terminator will CHANGE, depending on the eccentricity of the orbit. As the planet reaches it’s smallest orbital radius, the angular change will happen faster and faster, as the planet’s orbital velocity increases. Then it will slow after that point, until it reaches it’s furthest orbital radius, where is orbital velocity will be slowest.
Kepler's laws would be a good place to get a sense of this part. https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
Note that after a full planetary orbit, (one year), the planets terminator angle, will return to the original measurement, the same as excatly one year ago.
I have not actually tried figuring out the formula for all this; but I suspect this basic system of deduction would provide the eccentricity information you need (where they are in the elliptical orbit, how many months of winter to expect, etc..) by simply noting the changes in the angle of the planet’s terminator, each month.
This would NOT provide everything about the orbit, like the orbital radius. Nor does it account for things like, how long it takes for the moon's weather to warm and cool.