It’s way cooler.
Your dome would need to be strong enough to provide a useful amount of air pressure. If you’re willing to resort to a pure-O2 (but extremely dangerous environment), you only need to retain about 3 psi, and then to enclose an area of one square mile, your dome needs to be held down with anchors that can withstand 12 billion pounds of force. That’s a hefty and expensive dome.
In contrast, if you just submerge the entire moon in a sea of air, the weight of the air holds itself down and creates the needed pressure at the moon’s surface; you only need a gossamer layer of fabric above it all to hold down those few molecules of air at the top that have gone ballistic and are in danger of escaping the moon’s gravity altogether.
Sure, that’s easy and cheap, right? At least we have experience building domes and SCUBA gear. How are we going to manage enclosing a celestial body in fabric and air? And how do our rockets get in and out of the floating sheets above the atmosphere? What experience do we have with something like this, and how can you be sure it’s easier and/or cheaper?
We’re talking about radically altering the surface of a planetary scale object; “easy and cheap” are pretty much off the table.
We’ll likely just use a space elevator instead; with the Moon’s lower gravity even with modern materials we can build one there.
You could leave a gap. Here’s an image I’ve made of a paraterraformed world with a gap around the equator
There are other ways of doing it. To get even more living room on a Moon-like world you could dig massive underground caverns and fill them with atmosphere; the material excavated from the caverns could be used to build domes on the surface. On a geologically quiet, reasonably low gravity world you could dig deep and build high, ending up with multiple levels (or shells) of habitable chambers all protected from cosmic rays by layers of rock.
Actually, you can’t: The Moon doesn’t rotate fast enough for a space elevator. If you’ve got the cable material for a space elevator, though, there are other ways you could use that for cheap space access, some of which would work on the Moon.
This is the crux of the discussion - if you dump a large amount of gas or water on the moon, it should not “boil away” immediately. It’s simply that it would likely disappear with a half life measured in centuries, or millenia - basically “immediately” in the reference time of astronomy or geology, millions and billions of years. …which is why there is no significant atmosphere on the moon.
The other problem is that here are not a lot of sources floating around the solar system of nitrogen, the most convenient “inert” or filler gas for atmospheres; by contrast, there’s plenty of ice, methane, and other sources of H and O. Divert a few comets and icesteroids to hit the moon. Be sure to miss the earth, and not splash so badly it hits the earth.
The ideal moon launcher would be the one from Moon Is A Harsh Mistress, a maglev rail. with the low moon gravity, find a high peak and launch - get a good head start on orbit or escape even when there’s an atmosphere to contend with.