It will help to know the exact layout of the network.
What modem do you have? Make and model please.
What model TC?
How is it connected to the network and working in what mode? (bridge or router??)
What OS is on the MacBook?
I presume that the printer is connected by wireless or ethernet not USB?
I have restarted and updated both the Airport firmware and the printer drivers.
Did you reset the TC to factory and start over with a clean configuration?
I don't think your TC is at fault.. if you have Sierra on the laptop then maybe it is there.
Has this setup been working properly in the past, before Sierra and now not working since Sierra??
Sierra increased security to network drives.. in a way that bit back at lots of things..
The problem Sierra has is not actually being able to maintain the connection to the TC.. if the computer sleeps you need to re-authenticate to the network connection.. this is deliberate to stop ransomware wiping out your backup.. but unfortunately the bugs are still in the system. (It is also very hard to increase security without making itself a total pain).
Now for some suggestions..
1. Make sure you don't have any AV, security or adware on your Mac.. any of those 3rd party software programs can result in your connection being impossible .. not just difficult.
Apple specifically mentions AV software must not scan the destination drive.. but our experience is much more difficult and we recommend full removal if you hope to get TM to work.
2. The Mac must connect directly to the TC.. either wireless or better ethernet.. not via another network router or AP.
3. The Mac must have IPv6 set to link-local only.. not auto. (unless you are on IPv6 Internet).
4. Reset the TC to factory and use very short names.. Not apple recommended.. names <10 characters, no spaces and pure alphanumerics. Passwords same mixed case and numbers but >8 characters.
5. IP address does not change.. This is not an issue where the TC is main router.. but if it is bridged talk to us.
It can be set via the TC itself or via the main router DHCP reservation or use static IP setup instead of bridge.