Let me try. I'm not sure if I'm right so take everything with a grain of salt, but that's how I'd approach it.
Instead of trying to find the precise eclipses that will occur (based on LTAN) and trying to adapt to their duration, I'd choose a worst scenario eclipse: the satellite moving in the same direction as the Moon, as result staying in the shadow the longest, and also, with Moon at the perigee, resulting in largest umbra diameter and largest Moon linear velocity (again prolonging the eclipse).
It's some simple trigonometry to obtain the diameter of umbra and penumbra at Earth's terminator line distance, with Moon at perigee.
The rest of the calculations happen in the orbital plane of the satellite, or the plane of terminator, or plane normal to the Sun-Moon line, at Earth's distance - which one is pretty insignificant as they are all very close to each other and differing by cosine of angle between them -> minimal error. Let's assume they are all the same plane so that you don't need to e.g. angle the umbra cross-section.
Calculate the curve of the trajectory of the satellite (vs time) in the reference frame of the (moving) center of umbra in that plane.
Then you have your optimization task:
Calculate the density of penumbra, say, in range [0,1] 1=full sun, 0=full shadow.
It will be awfully nonlinear - decreasing from outer edge to umbra edge with surface of circle partially obscured by another circle (relative displacement of the circles proportional to distance through penumbra, diameters of the circles proportional to angular size of the Sun and the (Super-)Moon as seen from Earth respectively). And a flat zero through Umbra.
Multiply by solar irradiation constant. That way you obtain the irradiation density field. Again, multiply by panel efficiency and you have your power input density field. Subtract your power output (constant) from the whole field function, then discard the positive part (more input than output while the batteries are full, power wasted). You have a field of "power debt"; how much power your satellite draws at each point of eclipse.
And then you apply your trajectory (f(t)->x,y) , offset (translated) by parameters $x_0,y_0$ relative to the field center. Apply integral (dt) along the curve over the scalar field, and optimize the resulting function for such $x_0,y_0$ that the integral value is lowest (most negative). It's important to integrate over time, not just location as your velocity isn't constant.
The value of the integral is your energy output during eclipse - the value of energy that must be kept in batteries to "survive" between break even points between input from the solar panels and power requirements.
As per your edit: the integral over such a nonlinear field would be a total overkill for calculation by hand.
- curvature of the trajectory across penumbra = straight line. The diameter in shadow relative to the diameter of the orbit is small. This reduces the whole optimization task to crossing the diameter of the shadow.
- ballpark diameter of "debt circle" within penumbra as ratio of power input to power requirement. Say, if your solar batteries produce 20% more power than necessary, reduce penumbra diameter by 20%. That way you drop the awful task of calculating "surface of circle partially obscured by another circle".
- Assume all, umbra and penumbra within the "debt circle" is flat zero - solar batteries don't contribute anything.
- assume constant velocity. That way your energy requirement will be just "time spent in 'debt circle'" times power requirements of the satellite.
- penumbra and umbra size calculation
- satellite velocity in the umbra frame of reference if the Moon and the satellite move in the same direction.
- irradiation, panel efficiency, reducing size of penumbra to where panels don't suffice (although with much simplified function).
Explain what simplifications you've made. And also, that you'd want a bigger battery anyway, as a safety margin.