I have largely found the "miracle" of champagne yeast to be a myth. It hasn't worked well for me. The failings of pitching dried champagne yeast is primarily due to two things.
First, champagne yeast prefers to ferment the simpler sugars found in grape must. A stuck ferment in a malt based wort generally has a higher ration of complex sugars to the simple sugars. The beer yeast already fermented the easy stuff. Champagne yeast, therefore, will have a tough time with those sugars.
Second, a partially fermented wort has experienced a drop in pH and an increase in alcohol. Most any yeast, champagne or brewers, will have a tough time going from a dormant state (dried or liquid) to a fermentative state under those conditions.
I have seen that champagne yeast will get things moving again, but I have yet to see something ferment out with champagne yeast. It usually still will finish several points above where it total attenuation could have been.
A better approach is to get a 500ml to 1L starter going. Wait until it is very active visually and pitch that active starter in. Picking up a little oxygen at this point incidentally from pitching the starter won't hurt much.
You can do this with more of the same yeast already used, or chose a yeast with a higher alcohol tolerance than the one previously pitched. Before pitching the active starter I'd recommend warming the stalled beer up to 68F-70F if it isn't there already. (which is usually a better place to start to unstuck a ferment than pitching new yeast, IMO)