Connect an iPad Pro to both Ethernet and HDMI
Only thing I found through some searching around is this:

Lightning port to USBc adapter

to

Then USB-C Digital AV Multiport Adapter

Then get a USB to ethernet.
I am not sure if that Multiport adapter works well having the HDMI and USB port accessed at the same time or if the iPad will even utilize all the conversions. But there are examples of someone using a grocery cart of adapters to make something work on an iPad