I like aeration, and do it myself for minibogs, but not on a large scale. I've never tested it in a double-blind way. It may be useless.
That said, I have been advised that a thorough, "flood-it-out" watering with fresh rainwater once every month or two will do the job at little or no extra effort. Mother Nature will do this for you in many places. This makes a lot of sense to me.
If it's large enough, water will be the only reliable carrier of oxygen deep into the bog.
However, if that is the case, consider simply exposing the water to air as much as possible by creating a "pond" space of open water. Water breathes on its own. No pump needed, if the open area is shallow enough at one end and wide enough. You could let water plants grow there, or if you are me you would discourage them for ease of maintenance.
If the bog is small enough, I can tell you that direct aeration will do as well as circulating the water, without the mess. Essentially, you *are* aerating the water content of the soil, and in a small bog it tends to get sopping wet, so the gas gets transported pretty well. However, placement is important, as the majority of the air will travel back up along the outside of the tubing--it will try to push the soil away and form a path for itself. One way to avoid this to some extent, and to save on pump outlay, is to place a perforated plastic pipe in the bog that extends to the bottom and aerate the water that gathers in it. Another way is to run the tube, perforated at regular intervals, throughout the soil, terminating it at the very bottom.
Beware the accelerated drying mechanical aeration causes.
My two cents. YMMV. IANAE.
Steve