Is there any way to make the Recycle Bin invisible to all but a few individuals, or to hide its contents so that participants will not be able to restore files?
Unfortunately there is not a way to make the Recycle Bin, as a whole, inaccessible to students in a course room or to participants in a non-course room. However, you can keep a file from being restored by participants by first limiting the "who can open" access on it prior to deletion (or by moving it to a folder that had limited open access prior to deleting it).
If a file had restricted open access or had been within a restricted-open-access folder, such as the Instructor Folder (or a subfolder thereof) in a course room, and then the file is deleted, the deleted file's title will not be shown to students or participants browsing the Recycle Bin. The listing of files there "respects" the overall access control settings at the time of deletion, including restrictions on containing folders even when the file itself had no specific restrictions. The opportunity to restore a restricted-access file is never presented to those who did not have the right to open it in the first place, even if the file's owner and room coordinators can still see it sitting in the Recycle Bin.