Wouldn’t that be a _lunar_ heritage site, not a _space_ heritage site?

Well in theory it could even be done with the station keeping capable modules, a very gentle push indeed, it would just take time and maybe refueling..

Pretty sure you can ion thruster it if you are patient

If I remember correctly, a constant thrust of only 5 Newton (for comparison that’s the weight of a mass of 0.5kg on Earth) would be enough to counteract drag of the atmosphere on the ISS. That’s in the range of an electric propulsion systems like VASIMR that is under active development. Any higher thrust than that would automatically lead to orbit raising.

Progress can boost because it docks on Zvezda. Dragon doesn't, so you'd need a new docking method.

There was a proposal to use a VASIMR ion engine as an experiment to boost ISS, but that proposal was dropped a few years back. Had it been kept and worked then perhaps it could have worked, but it used a lot of electricity to do so

If the orbit is higher it can last for thousands or millions of years

Orbital decay comes from something. At some distance from Earth, you run out of somethings (most critically: atmosphere) decaying an orbit.

So past a certain distance, orbits are effectively "forever" ones.

It just takes more energy to get there, which is why the ISS is where it is.