I believe there's a 4k/120 HDMI switch available on Amazon for a reasonable price now ...
Yep, found one.
8K 60Hz HDMI 2.1 Switch 4x1 4K 120Hz 48Gbps,HDCP 2.3,ARC,VRR,CEC,HDMI Switcher 4 in 1 Out,4 Port HDMI Selector,IR Remote,3D,HDR 10,Dolby Atmos,for QLED TV,PS5,Xbox,Fire Stick,Roku,Blu...