I needed a bunch of fog on stage all at once (through a load-in door all the way US) at the top of the second act of “Tooth of Crime” several years ago. My solution was to start the fogger behind a closed load-in door several minutes before the fog was needed and let it run. Once the act started it was a matter of running up a set of PAR cans (blinders) focused at the audience, opening the door, and using a couple of stagehands (behind the lights) to fan the fog onstage with large (d-size, 24"x36") pieces of foam core. This should work with a cheap fogger. In my experience, they tend to cool down more quickly than the good ones, and thus produce less fog, so you'd have to experiment with timing.
Maybe you could devise a enclosed "stuffer box" that you could run fog or dry ice into with an in-line muffin fan and then release through another large vent at the appropriate moment. Size of box and length of time to compress the fog will vary.
The aerosol foggers you mention are a bit weak for this kind of work. There’s nothing better than a CO2 fire extinguisher for a blast of steam. The noise it makes is great too. It 'jets' rather than billows though. The company that services your fire extinguishers might even be willing to provide a loaner. If they have any waiting to be recharged they might be willing to let you discharge them first. We did a show with a CO2 effect a few years back and the
campus fire marshal let us raid his storeroom of extinguishers waiting to be sent off.
See if you can get hold of a couple or more old, won't be used again steam irons. Set them about mid range [may require experimenting]. Fill with fog juice. Allow to heat. For a brief, high density burst, there's nothing like a couple of these set to 'high steam' and turned to face down in front of a suitable fan. And it's cheap. Depending on length of run, you may have to shoot de-ionized water through them to clean your new 'foggers'.
I like the CO2 idea. It would not be difficult to build a manifold that would help disperse the cloud formed. The only problems I see is that the cloud might settle to the deck, and much of the output would be solid CO2. I'd thought of that, but I think we can work around it. The "explosion" happens just offstage, and the set design includes a window through which we can direct the "steam cloud" so it (a) looks cool and (b) keeps the frosty stuff out of the main traffic area for the ensuing scene.
Fog juice is mainly water, with additives to ensure that the steam created in the machine hangs around along time. Diluting it with distilled water reduces the persistence, and saves money too! For a show some years back, I needed the effect of a fractured heating pipe emitting a jet of steam. Diluting the juice ten to one with water worked great, but for the effect that Pat was after I suspect that five to one might be a better mix.