Go with CAT6 and you should be fine if the total length of all 3 cables is less than 100m. CAT5 will work for the TV but CAT6 will work for more uses in the future and shouldn't be too much more expensive unless your electrician has a bunch of CAT5 that they are trying to get rid of.
Edit: Also if you are thinking of wiring more locations the ideal solution is to bring all of the cables to a single location where you can connect to an Ethernet switch.
I've had a system in the late 90s with a 3dfx voodoo card. Also had a laptop with a SIS card from the early 2000 era.
The voodoo card was THE card to have it it's day (mine was an older second hand system though). The SIS card... for some reason they decided that standard VESA mode probing wasn't a thing they supported and would hardware crash when that API was used. I eventually got it working in Linux after patching xfree86 to not attempt probing when loading the VESA driver.