Now, on to problem number Two. A balloon can only be blown up a certain size before it pops. Your wall cavity won't pop, but you will be running a fan to build up pressure inside a cavity. It is a hole, a container, a space, that you want to receive air into. Either the chosen cavity is airtight or it is leaky. In one case, you will be running a fan against a buildup (air pressure buildup) which will cause the fan to simply maintain a head of pressure and not move anymore air at all after only seconds, not minutes. Not what you want at all. Air can only be compressed so far before it becomes a massive obstacle. Air is not infinitely compressible. Not at all.
In the other case, you will be sending air into more places than you appear to be planning for, because it will leak forward throughout everywhere it can. Dusty, inter-joist, dirty air, getting swirled around by fresh bathroom air, leaking out of switchplates and electrical plugs and door frames and Who Knows Where Else. -- Aghh... Disgusting. Hope this is clear. The dust left over from the initial construction is best left undisturbed. My opinion.
David