As I understand it, objects passing the event horizon of a black hole from their perspective will, from an external perspective, seem to keep approaching but never reach the event horizon. Given this, if one was to observe the gradual evaporation of a black hole, how would this affect the image of the object approaching the event horizon, leading up to the black hole's death?
What would an external observer see happen to objects entering an evaporating black hole?
Black holes evaporate constantly, but this evaporation is usually slow (relatively).
You are correct to say that an external observer will see the infalling (approaching the event horizon) observer as if that never reached the horizon. The black hole's stress-energy is so enourmus that (the difference of stress-energy near the event horizon and far away from it) cause a time dilation that is significant. This time dilation becomes infinite as the infalling object reaches the event horizon. The far away observer will see (if they could compare) as if the clock near the event horizon would stop as it approaches the event horizon. Thus the last image the far away observer will see of that infalling obserever is being frozen on the event horizon.
The infalling observer will not notice anything special on its own clock. It will seem to tick normally, as long as they don't compare it. In fact, the infalling observer will not even notice passing the event horizon (disregarding being spaghetiified). If they compare the clocks, though, the infalling observer will see the far away clock to tick extremely fast.
Now you are asking what the external observer would see if the hole was evaporating. Now the external observer's timeframe and time period is like let's say a 100 years. Now the external observer will be able to observe the hole for 100 years in their own far away frame. This 100 years as I explained, could be just a microsecond (or even infinitely small) in the frame of the infalling observer.
But in the external observer's frame this 100 years means that the hole will evaporate very little. Again, in the hole's frame this 100 years will be infinitely small period of time, so the hole will evaporate very little.
So to an external observer the hole could evaporate in an infinitely long period of time only. Thus the evaporation itself (regarding the infalling observer's image) will not change what the external observer will see.