Is it possible to transmit a decoder's UI over Wifi instead of using an HDMI cable?
Rather than use WiFi, you can use a specific transmission system.  These are known as videosenders or digisenders.  I use a German made system for sending the signal from my Sky satellite box to the television in the bedroom.

They are available in the 2.4GHz and 5GHz bands.  For the HD that you require I suggest that you would need to use the 5GHz variant.  You will also need to ensure that the version you buy has suitable input connections for the feed that you can provide.

All videosenders have a backchannel for sending signals from the remote control to the box downstairs. 

They are quite common in the UK, there is some information here but it refers mostly to the 2.4GHz variety.