We suppose an international company. That company buy bulks of cryogenic third stages from countries willing to sell them
The company evidently also buy a rocket ride to loft these stages into low earth orbit. Evidently no payload is carried; no GEO satellite.
OK, so far, this is pretty much like the CSI Lunar Express, so it sounds good at this point.
Meanwhile, lunar heavy payloads (15- 20 tons) are launched and dock to the ISS.
How exactly do these lunar heavy payloads dock with ISS, and what fraction of the injected mass is allocated to orbit matching with ISS, prox ops prop consumption, rendezvous gear, power for the rendezvous avionics, etc.?
The company provide the cryogenic stages with Soyuzautomated rendezvous and docking gear (Kurs + probe-and-drogue).
Then a booster loft the cryogenic third stage near the ISS no fly zone, the payload disengage from the space station, dock to the stage, fire, head to L1 / L2 / LLO.
This a mere expension of CSI Soyuz/ block D scheme that involve the ISS into a lunar flight.
By using the ISS for a lunar program, we don't have to wait 2028 and the end of its useful life to return beyond LEO...
Getting your hands on Kurs for the cryogenic upper stage will cost more money than you could ever imagine. Plus, your lunar heavy payload will require active Kurs to dock with the cryogenic stage, which means that the avionics of the lunar heavy payload would have to be "married" to the Kurs system - you can't just take any flight computer and hook it up to Kurs, its got to be designed for it, since Kurs handles much of the computations for docking.
Lunar Express makes lunar missions for human affordable, since it re-uses Soyuz, obviating the need to pay for Soyuz for lunar missions. Your architecture seems to make lunar missions more expensive, but for no obvious benefit other than "involving ISS".