As the title said; why couldn't Stargate Command use the Asgard cloning device to make more ZPMs, the way they did for the Replicator gun or the actual Replicator they created?
The fundamental issue with ZPMs is that they exist simultaneously inside of and outside of real space. As is clearly stated, the power source of a ZPM is basically a pocket dimension from which energy can be pulled.
Asgard cloning devices can undoubtedly produce the hardware required for a ZPM; the issue is that the creation/isolation/plotification of the pocket universe necessary to actually give the ZPM power requires more power/unobtainum/plotinium than the device can realistically output.
Long story short, it's an issue of a ZPM being more than the sum of its parts; cloning one wouldn't really do anyone any good.
From what we see in the series, a ZPM essentially consists of three parts:
A hull (likely the thick disc at the end of the ZPM), a mechanism with an interface and power flow controllers (likely the rest of what we see, including the crystals), and a power source (inside the hull, essentially consisting of a pocket universe from which zero-point energy can be drawn).
A matter replicator, only being able to replicate matter, would only be able to replicate the hull and the mechanism, and would most likely do so quite efficiently.
The sticky issue is that what you now have is a depleted ZPM. ZPMs are often described as batteries, but this is only half-true. They are like battery cells in the sense that they are a way to transport energy off-grid, but that's where the similarities end.
Battery cells derive their energy storage capabilities from differences in electric potential resulting from the oxidation states of the component chemicals, and the freeing of this is an electrochemical reaction of physical components.
ZPMs, however, derive their energy from subspace, and freeing this energy is an electronuclear reaction, involving no physical components to speak of.
To recharge a battery, you can simply shuffle the electrons around, rebuilding the electrochemical potential.
Recharging a ZPM, however, is a much more involved process. It's never explicitly discussed on screen, but the Asuran arc of Stargate Atlantis implies that creating new ZPMs is indeed possible, given the appropriate power generation capabilities.
We can infer that the process of recharging a ZPM involves creating a pocket universe and placing it, or a link to it, within the ZPM hull.
It is also possible that ZPM-grade pocket universes are primordial, but this is unlikely, given the difficulty of procuring a ZPM: Had these universes been primordial, recharging a ZPM would be a simple matter of finding a suitable universe, and creating a wormhole to it, a process that requires little more energy than dialing a Stargate, and might even require less, given the extremely small cross-section of the wormhole required (power requirements for Stargates are explicitly said to be exponentially related to the area of the gate in the Ori arc).
More likely, the creation of a universe from scratch is required, which given what we know about power required to operate the chair weapon and how it relates to our own power generation potential likely requires a quite monumental amount of power, likely on the order of several petawatt hours (zettajoule scale), i.e. something along the lines of the total world annual energy consumption, and possibly even several times that.
The Asgard Core is powerful, yes. It's able to freely reorder atoms and molecules, as well as providing cold fusion and fission for the purpose of coercing elemental composition, essentially turning air into rollerblades or a cello.
However, considering how it works with existing matter, the energy requirements of such operations pales in comparison with the energy required to hold back billions of tons of water for thousands of years. Hell, if a ZPM can hold back a supercell storm, the energy required for that alone is on the order of one tenth of the world annual energy consumption.
Most plausible in-universe explanation would be that Asgard cloning device cannot clone just about anything, and has certain limitations, just like replicators from Star Trek. Although advanced, Asgard technology is by no means limitless.
If you think about it, a device which can clone any object creates a logical paradox. I mean, if I had an Asgard cloning device with that kind of capability, the first thing I would create would be more Asgard cloning devices.
Of course, there are also all the possible real world explanations: they didn't think of that, ZPMs are made of plot etc...
While the Asgard were one of the most sophisticated of the races of the Stargate Universe, they were not able to replicate every technology of the Ancients. The Asgard Cloning Device was used for cloning living beings. Since the Asgard used genetic manipulation to create their bodies, having engineered sexual reproduction out of their biological processes. The cloning device was strictly for the creation of new bodies for the Asgard.
The device used for the replication or creation of existing technology is the Asgard Matter Conversion Technology. This technology could convert stored forms of information into matter creating from matter on their starships. This seem analogous to the replicator technology in the Star Trek franchise. The device seems perfectly capable of replicating anything it had stored within its database.
Seeing that the Zero Point Module (ZPM) was created by the Ancients, it is unlikely the Asgard would have one to reference. Given that the Asgard may have been able to find one at some point, having the plans for the device would not necessarily give then the ability to activate the process required to create the energy propagation powers of the ZPM. The technology of the ancients was as far beyond the Asgard as the Asgard were beyond Humanity. They understood the technology and could interface with it, but could not create it, even if they could replicate the hardware.
Consider this the same as creating a car engine without knowing how to create the gas required to power it. The physical device could be replicated without having knowledge of the process required to cause the ZPM to release energy. No other race has mastered the Zero Point energy in the Stargate Universe and the precise method of creating energy is still unknown. The Ancients created a technology that was user-friendly, relatively adaptable, but the manufacturing process remained only with the Asurans (Replicators) who were a product of the Ancients themselves.