There doesn't seem to be any windows 10 drivers for the motherboard, is this going to be why it can't be changed to 1000Mbps?
Have you checked your lan cable is capable of 1gb connections for the pc? If it’s not the connecting at anything better than 100mb the cable may not be able to do the 1gb speeds
What Cat6a?
You only need Cat5e for this. Cat6a will obviously work, but it's overkill.
NETGEAR Orbi Ultimate-Performance Whole Home Mesh Wi-Fi System - Wi-Fi Router and Single Satellite Extender with Speeds Up-to 3 Gbps Over 4,000 sq ft, AC3000 (RBK53) just a tad over your budget at £280 but worth considering.
I didn't think they'd do something like that. That's great news.