How do I shield PLB signals?
You require a Faraday cage. I suspect a PLB is small enough to use a test chamber for, which might make it easier.

You can create a suitable enclosure using a wooden frame covered with close mesh chicken-wire. Ideally, the DUT and the test equipment would be within that enclosure.

The output of the transmitter should be connected directly to a dummy load, of high quality, to reduce radiated leakage signals. Most RF communications test sets are built that way anyway.

Is there not an option to put the device into a test mode that reduces the output power/works on a discrete frequency/sends a 'TEST ONLY' message?

To be frank, since 406MHz is now the de facto distress frequency, monitored 24/7 worldwide by satellite, servicing them really is the domain of properly equipped RF workshops