According to page of that HDMI Switch that you are planning to buy, the HDMI version 1.4 is supported. The maximum speed of the data transfer over HDMI is given here, as 8.16 Gb/s. Notice that the speed is measured as Gigabits per second.
Now, what you want to render is 2560x1440 at 120hz. Calculating the bandwidth required:
2560 x 1440 x 120 x 24 = 10616832000 bits/s
This is approximately 10.62 Gbit/s (1 Gb/s = 1,000,000,000 bits/s). This is more than what HDMI v1.4 is said to be capable. So, it might not be possible to render at such a high fps.