For example I run a logitech media server, which is currently accessed at 'myhost.internal.lan:9000'. But I would really like to set up some way of accessing it with an alias such as myhost.internal.lan/lms
Depends on services and protocols - rmerlin's firmware does nice things, but it's limited to some degree...
nginx (or Apache with mod_proxy) can do this, along with some DNS records - and this is best done outside of AsusWRT - and you can still enjoy the trainwreck that AsusWRT has turned into... Eric has done great work here, but it's turning into a losing battle there with the upstream breaking things in the name of security?
DNS can basically be hosted anywhere - on the router, at the ISP, or in a box inside your network - to that end, an investment in something like an Raspberry Pi or other single board computer can extend the capability of your network.
Hosting DNS on the Pi(SBC) gives you the ability to define local DNS and Service Records - it also allows you to define hosts and services - with an HTTP service also hosted on that SBC, you can use proxy services to define port mapping across services (this is similar to what Google, Microsoft, and Amazon do with their cloud services).
These little SBC's don't cost much - RPi3 is $35USD, and it runs a big boy version of linux, not a variant of Linksys WRT54G drop from over 10 years ago...
This is a bit beyond what
@ColinTaylor understands or professes to know - and that's ok - he's probably smart at something, as he's been a great contributor in other threads.