Proper way to have software at a subdomain and a separate php socket?

OS type and version Debian 11
Virtualmin version 7.5

I wanted to install a piece of software and have a subdomain of an existing virtual server access it. Running php8.1-fpm.

I installed the software in a directory of the existing virtual server
Then > Webmin > Servers > Apache webserver > Create virtual host > Document Root = where I installed the software. > Server Name = what I wanted the subdomain to be.

Then created DNS master zone for it:

Webmin > Servers > BIND DNS Server > Create master zone > Domain name = the Server name above, > Master server = the existing virtual server

Then I had to fiddle with the apache config file newsubdomain.existing_virtual_server.conf to get SSL and <FilesMatch .php$>
SetHandler proxy:unix:/var/php-fpm/1672174475216688.sock|fcgi://

. . .

It kinda works. But it’s using the socket for the existing virtual server. And the url shows as existing_server/directory_of_installed_software (rather than subdomain.existing_virtual_server).

What would be a better way of having some software accessible via a subdomain of an existing virtual server, have its own socket, and have the url show as subdomain.existing_server?

I don’t think that a sub-server is what I want, since I do want the possibility to install the software not under the parent directory.

Or should I create a sub-server since that creates a separate php socket, then edit the apache conf to the Document Root that I want, and chown from the parent to the software’s user?