This might simply have been a confusion of units. Sizes.com claims that a Russian verst is equal to 500 sazheni, but a Moscovy verst is 1000 sazheni. The sazhen was fixed at 7 English feet (2.134m) in both systems way back during the 1700s in Peter the Great's rule.
In 397:1, the Aruch HaShulchan says that specifically, he is talking about the Russian verst (and therefore, in order to eliminate confusion and to inform heimisch readers how long the official verst is), he says 500 sazheni.
This perspective makes the end of 397:1 much clearer and less redundant:
ואלפים אמה הוא מיל, ולפי מדת מדינתינו רוסיא היא פרסה רוסית שקורין ווייארסט, שהם חמשה מאות סאזנעס. וכל סאזען הוא ד' אמות שבגמרא, כמו שבארנו בחושן משפט סימן רי"ח.
And two thousand amot is a mil, and according to the laws of our country, Russia, that is a "Russian parasa" which is called "verst", which are 500 sazheni. And all sazheni [i.e., Russian, Moscovy, Litovskaia] are 4 amot in the gemara, as we explained in Choshen Mishpat, siman 218.
When he talks about the distance to the moon, he says:
מ"ג אלף פרסה
We have an ambiguity there. He might be talking about the Russian "parasa" (2000 amot), the Moscovy "parasa" (4000 amot), or the Talmudic parasa (8000 amot). But all of these definitions are off. Even the Talmudic parasa would be, as @msh210 said:
43000 * 8000 amot = 43000 * 2000 sazheni = 43000 * 14000 ft. = 183,500 km
Perhaps, the Aruch HaShulchan heard that the moon is ~180,000 versts away. The speaker was speaking in Moscovy versts:
= 180,000 * 1000 sazheni = 180,000 * 213.4 m = 385,000 km
... but the Aruch HaShulchan understood 180,000 Russian versts
= 180,000 * 500 sazheni = 180,000 * 2000 amot = 180,000 mil = 45,000 Talmudic parasa
...which is just about right. Turns out, the moon's distance from the earth goes from around 363,000km to 405,000km, so we have some wiggle room.