If the escaped convict is reasonably capable of being arrested without the use of deadly force, then as mentioned above, the use of a drone to apply deadly force would be excessive force. This could result in judgments against the agencies and individuals responsible for the use of excessive force. Just because the escapee is sentenced to death doesn't authorize other people (even law enforcement) to administer that sentence.
It is interesting to note that there is one very close precedent for the use of a drone to apply deadly force by a law enforcement agency that I'm aware of in US History, which I've not seen brought up anywhere else here. On July 7 2016, Micah Johnson killed five police officers in Dallas and injured several more before barricading himself. Negotiations were attempted and deemed to be ineffective, so a decision was made to use a bomb disposal robot armed with C-4 to kill Johnson without further law enforcement casualties. To my knowledge, this is still the only instance of a robot or drone being used for deadly force by a law enforcement entity in the United States, and I've never seen a good argument for it being unreasonable under the extremely unusual and specific circumstances in that situation. There is quite reasonably plenty of concern about the precedent it set, but it's never been deemed excessive or unreasonable by the courts.
In terms of its relation to the original question, if the hypothetical escapee were barricaded and seemingly incapable of being arrested without further casualties, this precedent might be one that a law enforcement agency would point to when making the call to use deadly force, though it's certainly not legally settled ground. In that circumstance though, the escapee's death sentence wouldn't really be a justification for the use of deadly force.