I was able to resolve my issue and I figured I'd explain how I was able to get it to work in case anyone else stumbled upon this issue.
Firstly, I disabled Varnish since it seemed to be the source of my issues. I followed this guide:
I then went to /opt/bitnami/apps/magento/conf and changed my httpd-prefix.conf file so that the RootDirectory was "/opt/bitnami/apache2/htdocs", and my alias was directing to my store so
Alias "/store" /opt/bitnami/apps/magento/htdocs
After doing this I changed my base URL for Magento to reflect these changes- example.com -> example.com/store
Finally, I restarted apache, cleared my Magento cache, and all seems to be working fantastically.
I'm not sure that this is the right way--or the best way to accomplish this, but I don't see any issues.