This worked:
Update /etc/hosts with new line
10.137.110.241 rutx-wan
Then run
/etc/init.d/dnsmasq restart
Now Realtime Connections page shows rutx-wan.lan instead of IP.
NETWORK PROTOCOL SOURCE DESTINATION TRANSFER
IPV4 TCP rutx-wan.lan:42262 ec2-18-196-62-30.eu-central-1.compute.amazonaws.com:15010 4.18 MB (7945 Pkts.)
Separately have rutx.lan based on the System > Administration > Hostname being "rutx
root@RUTX:~# nslookup rutx.lan
Server: 127.0.0.1
Address: 127.0.0.1#53
Name: rutx.lan
Address 1: 192.168.1.1
Address 2: fda8:7132:98c8::1
root@RUTX:~# ping -c1 rutx.lan
PING rutx.lan (192.168.1.1): 56 data bytes
64 bytes from 192.168.1.1: seq=0 ttl=64 time=0.328 ms
--- rutx.lan ping statistics ---
1 packets transmitted, 1 packets received, 0% packet loss
round-trip min/avg/max = 0.328/0.328/0.328 ms
So now the question is how to make this automatic for any WAN assigned router IP? Maybe some tricks with /etc/dnsmasq.conf.