@pete g you only need the isolation transformer on the o'scope. Its only purpose is to prevent the oscilloscope chassis ground from interfering with the television chassis ground through a mechanism known as a "ground loop." Do not defeat the safety prong on the line input side of the isolation transformer.
The oscilloscope chassis ground is normally connected to the "green wire" prong of its power cord, as well as to the common or "ground" lead of the oscilloscope probe. Use an ohmmeter to check for continuity between the probe "ground" and the "green wire" prong. If any continuity exists you must float the chassis of the oscilloscope on the secondary side of the isolation transformer by defeating the oscilloscope's "green wire" prong with an adapter plug.
Once you do this, the "ground" lead of your oscilloscope probe, as well as the oscilloscope chassis, will "float" at whatever voltage you connect it to. Ideally, this will always be the circuit common of the television and that will be at earth ground potential by virtue of the "green wire" prong on the television power cord.
I hesitate to recommend connecting the "ground" lead of the oscilloscope probe to anything other than the "green wire" ground of the television set. Remember to avoid touching the oscilloscope chassis when it is "floated" by the isolation transformer lest you become part of the "ground" circuit.
It will do no harm to connect the DUT (the television) to an isolation transformer appropriately sized for the power drawn, but I can't imagine a troubleshooting scenario where it will help either. The whole point of using an isolation transformer on the oscilloscope is to isolate its chassis and probe "ground" from the power line ground. There can be a potential difference between "grounds" in the internal wiring of a building. If these "grounds" are connected together a ground loop current will flow that can and will interfere with any measurements you try to make, especially low-level signal measurements.
You "restore" the oscilloscope ground connection at the DUT when you connect the oscilloscope probe "ground" to a suitable common or ground point at the DUT. However, no ground-loop current flows because the isolation transformer interrupts the ground-to-ground path between the oscilloscope chassis and the DUT chassis.
Note that modern televisions use a coaxial signal feed from the antenna or cable modem to the television. The outer shield of this coaxial feed should be connected to a good earth ground, either by you or the company that installed the cable. It is this coaxial cable shield that should be be connected internally to the "green wire" safety ground of the television power cord, and eventually to the oscilloscope chassis via the "ground" lead on the oscilloscope probe.