Your only problem with the current transformer solution seems to be what happens to said transformer in the time between your TV going dead short (is that a frequent occurrence ?) and the fuse blowing (ought to be quite quick in the case of a dead short). If the current transformer primary is wound with something a good deal lower resistance than the fuse wire then it shouldn't ignite before the fuse opens.
However if this really is keeping you awake at night then could you protect the current transformer with a limiter across the primary ? Let's say it normally runs at 1.6A RMS (your telly's drawing 400W) and something over a volt RMS on the primary, and this will transform up to 24V RMS and 70mA RMS ojn the secondary to drive one of these
https://cpc.farnell.com/schrack-te-c...vac/dp/SW02586. The peak primary voltage should then be less than +/-2V.
If you took one of these bridge rectifiers
https://uk.farnell.com/vishay/pb3510...pb/dp/1815637# and cross-wired + to - then the two ~ terminals would each have two pairs of series diodes connecting them - one pair 'pointing' one way and one the other. This device passes no current in either direction until the voltage reaches about +/-1.4V. If it tries to go much above that then the current through the diodes rises very quickly and the device looks more and more like a short. It's rated for 35A continuously and 350A pulsed with a fusing I2t of 500A2s. That should be stronger than any reasonable mains fuse. If 1.4V isn't quite enough, given the 'normal' voltage across the current transformer, then you could cross-wire another bridge and put it in series with the first one. That would give you 2.1V, using just one diode pair in the second bridge, or 2.8V using both of them.
Cheers,
GJ