What is the effect of "very well" in the phrase
I can't very well talk to you and concentrate on sanding this at the same time.
The expression carries two or three effects:
Intensification: something like "absolutely can't".
Obviousness: bringing to the person's attention the fact that this is clearly not doable and they should have realized that.
Contradiction: highlighting the fact that what you can't do carries some inherent contradiction or impossibility.
You can't very well go to Australia when you're still bedridden.
Note that this doesn't appear to be the same use as the other one you identified, which I would say is primarily concessive or dismissive depending on tone.
I would venture to say that "very well" only appears with modals, and that the senses above are triggered by negative modals (won't, can't, couldn't, etc.).