I am not sure if Immich supports configuring base URL. You can look into that.

If it is not supported, you may need try your luck with URL rewriting. Nginx can do it natively. Briefly, you need to configure Nginx to strip the /immich from the request URL before proxying the request to the Immich instance.

Matrix’s official homeserver synapse is written in Python which is not known for resource efficiency. Adding in bridges does not make the situation look any brighter…

On the other hand, Pi Zero is powered by a single ARM6 core and merely 512MB RAM. ARM6 has been losing support due to its age and (lack of) performance. You may have a hard time setting the stage up. You may need to compile a bunch of packages from source which will likely take forever given the single ARM6 core. Also 512MB RAM is real small and you cannot have them all as the OS also needs RAM.

You can give it a try. But I bet it will be quite a torture.

Typical web servers like Apache and Nginx support virtual hosts (aka server blocks in Nginx terminology)