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.
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.