Luckily, I found this blog post that explained everything. The important part was a comment there that explained how to set a default encoding by adding the following line to my .profile file:
export __CF_USER_TEXT_ENCODING=0x1F5:0x8000100:0x8000100
While this fixed the problem while working in the Terminal, it did not fix the problem within AppleScript. I did find that I could add the aforementioned export command inside the do shell script, and everything seemed to work as expected. An example is shown below.
do shell script "export __CF_USER_TEXT_ENCODING=0x1F5:0x8000100:0x8000100; cat -s " & quoted form of (fileToCopy) & " | pbcopy"
