First of all, you will probably want to make observations when not illuminated by the Sun, so that the receiver is shielded by the mass of the Moon from the Sun interference.
Second, on the Moon there is no weather. You basically have only two conditions, it either is illuminated by the full sun or it is not (with some time while it is partly illuminated, but most of the time is just basically two temperatures). During night the temperature will be rather predictable for the entire length of the night.
So I think the simplest solution would be to just use the telescope for about 50% of the time when its geometry is stable and is shielded from the Sun.
It is not such a huge issue, the basic fact of this kind of telescope is that you can't steer it so you still rely on orbital mechanics of the body on which you place to move the part of the sky which it listens to.
In this case even restricting to 50% of the time does not change much as it still requires 1 whole year to make observation of entire part of the sky it can observe.
The changes in temperature are exaggerated. If you cover the telescope during day with some type of cover (like space blanket [1]), then everything that's not in direct light will stay at the same temperature as during the lunar night. For the simple reason that there's no air on the moon to transfer heat, and the regolith is a very good insulator.
So you could end up using the telescope 14 days out of 28 and be done with it.
But what's more, you can set up the space blanket to only protect the part of the telescope that's in direct sunlight. For the rest of the telescope it will appear like it's perfect night, because there's not atmosphere to disperse the rays of the sun. So you can end up using the telescope during the lunar day as well, although not at full exposure.
Out of 29 and a half actually.
I'm curious where the misconception that a lunar month is 28 days comes from. I believed it too but I can't find any good reason for doing so.
Separately though, the misconception comes from the fact that we have 4 phases of the Moon (new, first quarter, full, last quarter), and it's easy to conceptualize that the interval between two is 7 days, or one week, rather than 7.38264725.
The dust also picks up charge and levitates, but I suppose that allows you to sweep it up using a charged broom.
Well, yes, but at radio wavelengths, it would take an awful lot of dust to significantly impact the performance of the telescope. My gut says centuries worth if not millennia. And the article is proposing a wire mesh primary reflector, so dust will mostly just fall through.
Another issue is the change in temperature.
Similar answer to the above. The wire mesh reflector would expand and contract. I suspect the main impact would be to change the location of the focal point. We can handle that by moving the receiver up and down similar to what we do with telescopes on Earth.