I will say that I tend to rail against contractors wiring for home theater because of the quality of the installs.
Why not wire it fully to the TV with a few runs of CAT-5 as well as the HDMI cable and perhaps a piece of coax? Even if today it is a basic installation, at some point they may add surround sound, or want additional features which are not currently part of the setup, and then they can't get behind the walls to run them without tearing up drywall.
Obviously, if it is a straight shot down 3 feet, then no big deal.
But ANY installation where you are travelling across a stud of any sort should include a full A/V installation including HDMI, RG-6, three pieces of CAT cabling... Or, one cat-5, one HDMI, and a piece of 1.25" flexible conduit.
It's great that you can run the cabling for a client, but you can earn a bit more if you properly wire and then they won't be upset when they need that added capability which you never even offerred to them. If you OFFER a complete run of HDMI, Cat, and conduit for an extra 100 bucks or so, and they turn you down, then they need it, at least it isn't your fault for not giving them that option in the first place.
I'm not sure if any HDMI device is using ethernet over HDMI at this time.
Thin cable can be fine for short distances, but cheaper cables are cheaper and can be more prone to damage, especially at the heads where it connects to the main cable. There should always be some sort of redundant plan in place in case a cable does not work as expected.