I forgot to put the name of the MOSFET... post updatedYou may want to look into the gate voltage of your chosen MOSFET. This even if you consider a logic level MOSFET. There are switches designed around switching a HDMI signal. A simple Google of HDMI switching circuits will give you an idea of what's involved. Consider something along these lines.
Ron
Thanks for the answer.OK and this is your data sheet. Note the VGS of the MOSFET and your 3.3 V gate voltage. This is not a good idea. They make switches for what you are looking to do, I linked to one.
Ron
Yes, but I'm interested in understanding the reasoning behind using a MOSFET in this configuration.You could also likely get by using a simple reed relay as a switch.
Ron