If it's cheap enough, this is probably the way I'll go. Even if I only get 6-7 Gbps, that's good enough for me. This computer doesn't need to be as fast as my desktop, which is on 40GbEPCIE3.0 x1 is 8Gbps which is obviously less than needed for 10GbE. There are some rare x1 5GbE cards based on AQN108 chipset but if you would manage to find them in stock - they are listed at $200+ which defeats the purpose.
So just cut the slot (or use the extension cable) and find a regular x4 10GbE card which would work through a single lane. But be sure to pick a PCIE 3.0 one to make the most of that x1.
You probably know this already, but manufacturers have been playing that "wink and nod" game for many years. It was especially bad with early quad-port cards. Sometimes the sales guy would try to explain it away by saying you'd never be running at 100% on all four ports...PCIE3.0 x1 is 8Gbps which is obviously less than needed for 10GbE.