If I were to save the .CLASS ByteCode file in the home directory of the SD card, would the HTML code be able to access it directly?
Do you understand how a browser works? Specifically, have you ever looked at how a browser fetches a page? There is not a single GET request made. Instead, the main page is fetched, using a GET request. Then, that data is scanned for aref tags and linked files. Separate requests are made for each file. Similarly,
It is up to you to write code, on the Arduino, to parse the GET requests, and return the proper file.
Often, that file is simply constructed on-the-fly. But, it need not be. It could be fetched from the SD card, instead. But, the browser has no idea where the requested data comes from. It is simply a stream of data that the server sends.
So, yes, the .CLASS file can be stored on the SD card. But, no, the browser can't access it directly.