I downloaded and installed Affinity Designer 1.7 Beta. Pleased to say that this does NOT display the "The software requires a DirectX 10 compatible graphics card in order to run.." dialog.
I also have Affinity Photo 184.108.40.206 installed on my machine. This also works perfectly and does not display the DirectX10 dialog.
Oddly - I just re-ran Affinity Designer 220.127.116.11 and it also loaded perfectly without the DirectX10 dialog! Something changed behind the scenes! Perhaps a driver update or the Beta install etc.