In the air port utility it shows fair when in bridge mode and half duplex.
Can you do a screenshot to show me that. I have checked the utility but I cannot find anything that shows the status of a port as half duplex.
Any switch should default to full duplex. That is what switch means vs hubs. Hubs were half duplex devices and switches were full duplex. Any modern device with a switch chip is going to default to full duplex .. but it will negotiate the port speed and duplex nature of connection with the client it is connected to. Sometimes it can get it wrong..
It sounds like in your case a typical issue of the AC model which has negotiated wrongly.
While my older AirPort Extreme show full duplex in the airport utility, the new AirPort Extreme shows half duplex. Both airport extreme's were plugged into the same Ethernet cable.
That seems to be a bug in the new Extreme.. as Bob has noted use a switch in between your router and the TC and it might negotiate it correctly or use a crossover cable..
Before I get loud howls of protest.. this bug has been seen and this is pure pragmatism.. You do not need in this day and age crossover cables.. except they fix the bug. At least for some people.
See
Crossover cable to fix issue of connection to ONT or cable modem.
https://discussions.apple.com/thread/5391254?answerId=23236139022#23236139022&ac_cid=tw123456#23236139
I suspect Apple have a driver or actual hardware problem in the new AC version routers.
And so far .. it only affects the WAN port.. other ports are not being reported as having issues..
The best way to tell is use a managed switch.
I have a managed switch on my network which shows the speeds and duplex nature of all the ports.. these are not too expensive now.

The TC is on port 13 of the switch.
And I can quickly test all 4 ports .. all of which negotiate correctly to 1G Full..
But then I have a Gen4 not Gen5 TC.