Explanation:

- Warren Hastings: He was the first Governor-General of Bengal from 1773 to 1785. His tenure ended before the 3rd Anglo-Mysore War, which took place from 1789 to 1792. So, he was not the Governor-General during this time.

- John Macpherson: He was the acting Governor-General from 1785 to 1786. His role was temporary, and he was not in office during the 3rd Anglo-Mysore War.

- 2nd Earl Cornwallis: He was the Governor-General of Bengal from 1786 to 1793. During his tenure, the 3rd Anglo-Mysore War occurred. Therefore, he was the Governor-General during the war.

- John Shore: He served as Governor-General from 1793 to 1798, after the 3rd Anglo-Mysore War had concluded.