Letter writer snapped at my reminder email. What can I do in the future to prevent this, and should I still accept his letter?
How can I prevent this from happening in the future?


Well, you can always go live on a desert island, or join an order of silent monks (and even living among monks would not guarantee that you will escape such foibles of human nature).

More seriously: you can’t prevent it, and you shouldn’t try either. You did absolutely nothing wrong by sending a reminder email. For every professor who responds irritably to your emails there are at least ten others who would appreciate your reminder email, and for some of them the reminder would even be needed to get them to upload their letter. Thus, you stand to lose much more than you will gain by changing your behavior because of the professor’s response. Just accept that working with other people necessitates occasionally having some strange and perhaps unpleasant interactions (trust me, you will have many more), and move on. 

By the way, I also think it’s highly unlikely that the professor will modify his recommendation because of your email — that would be an extremely unprofessional and unethical thing to do, much more so than replying irritably to an email. Anyway, good luck with your applications.