Neither a Cat6 or Cat5e cable would really help as the BT socket will be an RJ11 connector and the cat5/6 cable would be RJ45.
You either need to put the router next to the master socket and then use the cat5/6 cable to route to a switch or device not in the loft, or you would basically want one of these running from the socket to the router downstairs.
https://www.amazon.co.uk/PCSL-Cable-Modem-Broadband-Connections/dp/B003JOJSJS
Personally I would stick the router in the loft, turn off the wifi on it, then run a cat5/6 cable to a dedicated WAP elsewhere in the property.