Gee, it seems that Louis Pasteur who was born some twenty five years after your Lord Amherst died, stole all that knowledge about germs and such that Lord Amherst had at his fingertips. I don’t know how the smallpox blankets deal could be true if it took another fifty or sixty years before anyone even thought about germ theory.
Before anyone of letters wrote about it. Considering that a theory had better be very defensible before being presented, it is difficult to ascertain whether anyone had thought of germ theory prior to it being presented and accepted.
However, you don't have to be a genius to figure out if you give blankets that belonged to people who got sick and died to people who subsequently get the same sickness and die, something in the blankets will make people sick.
Edward Jenner invented the smallpox vaccine in 1798, the year after Lord Jeff died and 24 years before Pasteur was born.
You didn't need the germ theory to observe that smallpox was communicable. Nor did you need it to observe that persons who had had cowpox tended not to get smallpox. Of course, it did take the likes of Jenner to figure out that you could protect people from smallpox by deliberately exposing them to cowpox.