Sure, cat6a is more than enough for this purpose: most products even work over cat5. The reason I suggested cat7 is to be future-proof if you want to use it for 4K or 8K in the future.
The ones I used are these: http://www.marmitek.com/en/product-details/watch/av-over-cat5/megaview-80.php, but there are a lot of brands that make these products. Some also transmit IR back to the source, which can be useful if you want to use an IR remote control (eg http://www.hdconnectivity.com/extenders/hdmi-over-cat5-cat6-with-ir-extender.html).
Of course 7m isn't a very long distance: a good HDMI cable can easily do this distance. I also don't think over ethernet will be cheaper, if you count the cost of the adapters + the cat6 cable. You also make your installation more complex and add 2 potential points of failure. You will have to decide for yourself if the advantages are worth it...