OK, I found how to solve my original problem and then ran into a different problem. Let me tell you how I solved the first problem to help other users that may encounter it.
In my media library, there are two Apply buttons next to different dropdown menus. Even though both of these menus contain the "Rename" option for your plugin, they have some other options that differ. In order for the bulk option process to work, I need to choose "Rename" in each of the two dropdown menus and then press the "Apply" button in one of them. Otherwise, the loaded won't appear next to each "New Filename."
Now, here's the new problem...
Even though the bulk process changes all of the filenames of the images I am trying to rename, only some of the links to these images are updated in the posts that contain these images. For example, if I do a bulk rename for image1, image2, image3 and image4 and all of these images are included in a post, maybe only 1 or 2 of the links to the images will be updated. The rest of the images links on the post remain pointing to their older filename. Why is this? Any ideas how I can fix this?
Otherwise, I have to go to the posts that contain the images and edit each link that didn't update manually, which defeats the purpose of the plugin.
By the way, I just made a donation to your plugin via PayPal.