This is fairly regular issue.
There are a number of things you can do to HELP the problem but nothing will 100% fix it.
Fact is the Apple Bonjour is not particularly great with many routers handling it poorly.
I have same kind of setup.. Asus main router.. TC plugged into it but these are the settings that help.
1/ Turn IPv6 to link-local only. This is under internet options.

2/ Set a static IP.
On the Internet tab.. it will be set to DHCP.. change this to static.
You must manually add a DNS.. i.e. same as router or a public DNS.. whatever..



My values will be different to yours.. as long as you record the actual IP and put it on a sticker on the TC so you don't forget it.
3/ If you continue to have issues mount the hard disk of the TC in Finder using Go, Connect to Server using IP address not name.
afp://192.168.20.212 (replace with your actual IP address).
Then reconfigure Time Machine to use the TC via its IP address not the network name.
4/ On the computer open your preferences to network.. I am using wifi but in your case ethernet.
Make sure domain is set correctly. It should be blank or local. It will give major issues if your ISP passes a domain prefix to your computer. This one is a bit harder to do but I also set the local domain in the main router to local (many routers by default will be lan or blank).

Now I would like to report everything is now wonderful.. but outside the Apple walled garden the world is a messy place. Things still occasionally force me to do a TC reboot.
