You want the TC to 'see' the AE network. Have you tried the TP-Link with the cable end connected to the AE and the other end somewhere near your preferred TC location?
The Apple solution would be to use an Airport Extreme as your network provider instead of the HH. There are a number of advantages in terms of coverage, speed, 5GHz network, guest network etc, some of which may be available on the latest HH - with which I am not familiar.
I think you can connect an external hard disk to the usb in the HH (with the cable leading to your underground security bunker) but I don't know how well that would work with a mac and time machine. There must be other wireless hard disc (or NAS) options - again, not sure how well they work with TM.
Last edited by Background Noise; 24th Mar 2016 at 11:12.