The experts may have a better way, however there is an Apache option that would work.
You can use ProxyPass to set up a reverse proxy from either a sub-directory or a sub-domain. Here is the web page on this: mod_proxy - Apache HTTP Server Version 2.4. See Basic Examples.
I have used this option on a server with several virtual-sites installed and it worked a treat!
I do have to warn you – MAKE SURE your web server is properly SECURED. Here are some security tips: Security Tips - Apache HTTP Server Version 2.4. You can find plenty of other suggestion searching for “secure apache”
Cheers!!