I think part of the solution will involve using "regular stuff" (which can be sourced from lunar atoms) like batteries.
Don't use lithium batteries to continuously power an oxygen generator. Instead, have an enlarged oxygen generator and store half the flow for use at night.
Don't use lithium batteries to power a heater at night, instead "pre-charge" heat in a dirt thermal mass during the day.
Don't use lithium batteries to power a continuous CO2 scrubber, instead use a liquid amine scrubber and recharge the fluid tanks during the day.
Etc etc
Obviously you need to do in-depth tradeoff math for each case, but my napkins say "the system is the battery" wins in many cases.
Storing 14+ days worth of <everything> is going to mean massive amounts of storage.
...which is why you want to be extra careful in how mass-efficiently you're storing it.
A tank of liquid oxygen masses only a few percent of its contents, and its contents store more chemical energy per mass than a lithium battery. Later on, insulated tanks are vastly easier to produce from lunar materials vs batteries. These are all nice leverage multipliers.
Nuclear is a popular notion, but the real-world economics don't favor it. Effectiveness beats elegance.[0]
Making liquid oxygen (as opposed to gaseous oxygen) is not a terribly efficient process.
You have to do the trade between liquefaction and storing oxygen as a pressurized gas. Reusing a single Starship's propellant tanks, for example, at 6 bar would hold ~14 metric tons of oxygen, enabling a continuous lunar population of over 1,000 people.
The compression or liquefaction energy isn't a show-stopper necessarily, it just works into the total energy efficiency of the storage system. Compression is nice because practical isothermal compressors are now being demonstrated.[0]
Both liquefaction and compression require cooling. I do think it probably makes sense to do the thermal storage trick in reverse also, storing up a bank of "cold" at night to be used during the day. The thermal radiator panels still provide (derated) cooling in the day, but the thermal storage pool time-shifts the natural oversupply of cooling power at night.
[0] https://newatlas.com/good-thinking/carnot-centrifugal-air-co...
Care to elaborate? As an interested layperson I always thought the major factor for losses in open air was radiation into space. As such an atmosphere's thermal mass would be benfitial, no?
A tactic I've used in Dyson Sphere Program is to build an equatorial belt around the entire moon so that some % of panels are always in sunlight. On our actual moon that would be approximately 7.2m panels, assuming they were a single tightly packed row of 5' x 3' panels.
This sounds like an insane amount but several of the largest solar installations on Earth each have more than that, with Bhadla Solar Park in India having over 10 million[1]. It's a dumb solution but is one which requires no shipment of energy storage. If you're able to mass manufacture panels in-situ, it may be simpler to do something like that than to ship large batteries to the Moon in order to use them as storage.
[1]: https://www.aljazeera.com/gallery/2021/11/3/india-solar-rene...
Oh, of course this is terrible now. It would have a very, very large required scale to come out ahead compared to just shipping power or power generation some other way (if this is true at any scale). It is also surely not fully baked yet.
On the other hand... it's a notable step towards being able to build "big" in space and might be worth doing to learn more about the practical aspect even if it's higher risk and more costly than shipping a bunch of PV.
The bigger problem is how practical is it to use PV at all on much of the moon.