Okay, having finished the book (again), I'm still not sure how compatible it is. The main problems that I see are:
1) The US having a cannon that can shoot a three-man capsule into Lunar orbit, way back in the late 1800's. The "shell" doesn't ever actually reach the moon (and in Around the Moon, it circles back to Earth to splash-land in the Pacific), but with better aim it certainly could have.
2) Being shot out of a cannon at 25,000 mph is a non-fatal experience as long as you use some water to absorb the shock. Similarly, landing from orbit is workable with 1870s tech.
The second point *could* be handwaved away, if we assume that IS physics are noticeably different from ours. Which they are, or ether props and cavorite wouldn't work. So maybe it's not a problem...although orbital reentry in an aluminum shell seems a bit of a stretch even then.
If we assume that the physics assumptions involved are correct, the question becomes "why didn't the US continue to launch moon shots until they succeeded?" They clearly didn't in the IS timeline, and that's hard to explain unless something went disastrously wrong somewhere. Maybe the "Columbiad" cracked after one firing, or worse, failed catastrophically on the second. A "launch failure" that blew Stony Hill and the city of Tampa to fragments would be an event on par with Vesuvius or Hiroshima. That (along with the original miss) might be enough to cool the "moon fever" the world was suffering from, at least until Cavor comes along.
So...maybe it works. The Baltimore Gun Club might be a valid IS organization, and if they are, the inheritors of Barbican and Marston would undoubtedly be involved with the US ethership program. And the BGC really, REALLY likes things that go boom.
Rich