OK. Here's how Kirk Prime would've done it. Easy peasy, and not at all as entertaining. Remember, no transporters.
said, use the cover of darkness. Send a shuttle down from orbit. The shuttle is reinforced to withstand the heat being put out by the volcano. Scotty has even added extra thrusters and stabilizers to cope with any turbulence.
Once in place, Spock uses his tricorder to scan the maw of the volcano for an appropriate spot to deliver the freezing device. Once found, the device is attached to a probe and delivered to the spot by Spock using a fly-by-wire system. Once there, the timer is activated, the shuttle leaves, and the device detonates.
That requires only two concessions: I would think 23rd century technology would be good enough for a fly-by-wire system to deliver the device (considering we have it now), and also good enough for activating devices by remote control from short distances even inside a volcano (considering we can already activate devices on other planets remotely).
Now let's say that took nine minutes. Which nine minutes would you like to see? That, or the original?