How to detect/measure a direct lightning strike

In modern times, I think that the least expensive way to do that is with a camera. A photodiode circuit can trigger the camera.

My company used to do triggered lightning research. We would launch a rocket trailing a wire, and take pictures of the resulting lightning strike (or multiple strikes). The picture illustrates.

Your research follows in the footsteps of McEachron, Vonnegut and others, ca. 1940. The Empire State Building in New York City was instrumented to measure the numerous, usually harmless, lightning strikes. Almost all the energy is conducted directly to the ground. (However, occasionally radio and television stations at that site were knocked off the air by lightning, since the antennae could not be absolutely grounded.)

Consider a steel-framed skyscraper, as your tower, is a fairly conductive structure. If the tower were a uniform width, 100 m tall, and you were to measure the voltage across a 100 mm (10 cm) tall segment, it would be 1/1,000 the voltage of a lightning strike.

Of course, there are inductive effects that could produce a huge current in a parallel conductor, but properly-designed apparatus placed within the tower footprint should be reasonably safe, since the tower is effectively a Faraday shield.

So certainly consider investigating. A high-speed, grounded or completely non-conducting optical cable could be used to carry data from the tower, or data could be recorded from safely inside the structure for review later, which might be best, since you only plan to review it occasionally. However, do consider the equipment may have a short service life, and value it as disposable.

As a rough outline, you might measure off a distance on a vertical support, attach two bolts, and run that voltage divider to a measurement circuit that would start collecting data when the voltage exceeds some limit (perhaps a volt or so). There are various circuits to cache high-speed data and to record it at lower speed, such as this from NASA, or in this paper by Grebovic et al.

If you have a genuine local lightning strike, the RF will light up a fluorescent tube. All you need to do, is put one in a dark box, and monitor it for flashes. You won't know exactly HOW near, but if a disconnected lamp lights up, there had to be an electrical event. Grounding one end of the lamp is optional, if my microwave oven experiments are any guide.

perhaps tape a pencil to the tip and wait for it to disappear?

A Rogowski coil could work - it's an air-core current transformer - so low inductance. but you'd need to isolate it well.

With the strong magnetic fields involved even a bare reed switch or a bare hall sensor might work.

Everything I read about Faraday cages tells me there's nothing detectable inside even during a lightning strike. But I'm willing to bet that metal cabinet is not quite textbook perfect, and as such you would expect to see some induced currents inside. They would be significantly attenuated compared to anything you might attempt to observe on the outside. This opens up the possibility of using instrumentation without the expectation of it being vaporized on the first hit.

I'm also drawn to the idea of a microphone, but we really should think more in terms of a vibration sensor (and a not very sensitive one at that) that can deal with the absurd sound pressure levels. Again, that metal cabinet probably makes for a nice stiff diaphragm of sorts.