If you have freedom to switch from Apache to lighttpd, then the most straightforward solution would be to use mod_secdownload which would do exactly what you want, that is, provide application authentication while serving the actual files via web server.
However if you are stuck with Apache, then I suggest mod_auth_token, here they mention PHP but you can generate the token in Python or any other language. Using mod_auth_token you will be able to generate the token in your application, and then have web server serve the static file utilizing that token.
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