good idea, but a slight correction - mDNS and SSDP are entirely different things, rather SSDP was Microsoft’s initial proprietary take on the idea, and mDNS was created as the new and improved standard.
mDNS does multicast (send/receive) on 224.0.0.251 and ff02::fb on port 5353, while SSDP uses 239.255.255.250 on port 1900 as you mentioned.
But I think OP’s issue is that they’re on different subnets; mDNS expects the server and the client to have a perfectly overlapping subnets inside the same LAN. If the server has 10.0.0.3/16 and 192.168.1.3/24 then the client must also have 10.0.0.7/16 and 192.168.1.7/24. Or, if you can tell the server software exactly which IPs to announce, then that might work too.
there are workarounds to this, using avahi reflector, but that thing is buggy – specifically you need to disable NSEC on the server, and lock it to either IPv4 or IPv6.
I made some note on additional pitfalls while i was writing my own mDNS and SSDP servers for fun, they’re at the bottom of this page: https://ocv.me/copyparty/helptext.html
Alpine also has the advantage of musl, which is a safer alternative to glibc, at the cost of some performance. So, if anything, I’d expect people to consider alternatives to Alpine for that reason, as alpine is already the best choice for security.