I recalled how Rembrandt's 'The Nightwatch' has been horribly vandalized on multiple occasions (seriously, wtf?), and yet somehow experts have been able to piece it back together. I saw it in person last November and there was no trace of damage that I could see.
So I knew there must be a way. Or rather, if I had the will, there would be a way.
In the past few days I have created a canvas patch with a primed-canvas inlay to perfectly fill the hole, and subsequently painted it over to blend it into the piece. I plan to paint a bird in flight over the area to further disguise any surface imperfections, and also think that this will add to the overall composition of the piece. Fingers crossed that all this effort will save the painting, although I imagine that it's value is significantly decreased due to the innate structural damage, no matter how well I conceal it. I'm not too concerned about that though, my focus at this point is on simply saving it.
This is the painting in question (this photo is pre-burn-hole. I have photoshopped a bird into the image, in the place where the patched hole currently is):