Any tips and pointers would be much appreciated, in terms of common ways to search for and exploit vulnerabilities, or the exploitation of other televisions. Alternatively, if this is an absolute fool's errand, and the whole thing is locked down tighter than Fort Knox, a firm warning that this is not a good thing to dick around with over a weekend would also be appreciated.
Its always best to work out where the mains PSU is, and either cover/isolate it so that you don't touch it by accident. Even if turned off, there is a non-insigificant risk that there is either mains power lying around or some other large charge/current ready to bite.
THe back of the inlet is also tends to be a hazard, so be really careful around there too.
[1] there are exceptions, e.g. older vacuum tube sets which used a mains-driven EHT circuit with discrete capacitors
The other answer is "pliers."
(Make a connection across the terminals that have a voltage difference)
That kind of tester is usually built into a screwdriver and often comes free with some electrical items. They are maybe $2 otherwise.