TL-WR710N can be configured as WISP Client router mode. You can use it to connect public hotspots. But the problem is whether your BT account can be used for BT Fon WI-Fi.

[FAQ ID: 617] How to configure the WISP Client Router Mode on the TL-WR710N
http://www.tp-link.com/en/article/?faqid=617

[FAQ ID: 614] How to configure the Client Mode on the TL-WR710N
http://www.tp-link.com/en/article/?faqid=614

Thanks, I've read those but I don't understand what settings to use in the wisp client mode.

Currently, I set the wireless settings to dynamic ip and when I try to connect to the internet, I get the BT Fon login page. So I enter my username and password and this gives me temporary access.

I'd like to automate this process, if possible.

Your current configuration is correct. That is how it should work

The router cannot complete the web authentication by itself because it has no web browsers, as a result you have to enter the information and finish the login on your laptop/smartphone.