What model TC from the base.. A1xxx??
How old is it?
Does the connectivity led come on when you plug the amp ethernet into the TC? Does the amp have ethernet connectivity indicator.. does it show speed as well.
It can seem strange that in this day and age ethernet doesn't work.. but it happens occasionally .. especially with a less than perfect cable. Sometimes locking the speed of the port helps but TC doesn't have the ability to do it, and I doubt the amp would bother having those sorts of controls.. but if it does, set specific speed, eg 100mbps full duplex rather than auto.
Another solution is stick a small switch between the two devices.. that often sorts out issues.
DHCP standards can change and get messy.
You should also assume anything outside the apple world will not take kindly to typical apple names.
I would reset the TC and use all SMB standard names.. short, no spaces, pure alphanumeric.. for everything.. TC itself, wireless and hard disk.