If it's connected to a tandem breaker (i.e. two independent half-width breakers in a single package), and you don't have any additional slots in the breaker panel, you won't be able to replace the breaker with a GFCI breaker. Otherwise, I also like this suggestion a lot if it's doable.
Moving the GFCI to the next outlet in the circuit would be a reasonable alternative provided the TV is mounted someplace that doesn't require GFCI protection. If it's in an outside bay, for instance, it should have a GFCI as that's a (potentially) damp or wet location.
A third option that may be viable would be to replace the TV outlet with a normal outlet and add a GFCI somewhere before it in the circuit, perhaps close to the electric panel, where it's at least halfway accessible to reset.